SOLUTION: 1. 2x-y=1 6x-3y=3 2. y=5x-1 y=5x+4 3. x+y=38 x=2y-25 4. -2x+5y=9 x+y=13 5. 2x+y=7 3x-y=3 6. y=3x+2 2x+y=-8 7.y=-x-3 y=-2x-8 8. y<_

Algebra ->  Systems-of-equations -> SOLUTION: 1. 2x-y=1 6x-3y=3 2. y=5x-1 y=5x+4 3. x+y=38 x=2y-25 4. -2x+5y=9 x+y=13 5. 2x+y=7 3x-y=3 6. y=3x+2 2x+y=-8 7.y=-x-3 y=-2x-8 8. y<_      Log On


   

Question 104244: 1. 2x-y=1
6x-3y=3
2. y=5x-1
y=5x+4
3. x+y=38
x=2y-25
4. -2x+5y=9
x+y=13
5. 2x+y=7
3x-y=3
6. y=3x+2
2x+y=-8
7.y=-x-3
y=-2x-8
8. y<_3x-2
y>x+1
9. y=x
y=2x-4
10. -2x+y=6
y=3x+9
11. y<3x-2
-3x+y-2
12. x+y=8
x-y=10
13. y=x-3
y=-3x+25
14. 4x=2y+1
2x+y=4

15. x=2y=10
2x+4y=10
16. y=x+4
y=4x+1
17. y=6x-4
y=-2+28
18. 3x-1oy=-25
4x=40y=20
19. 8x=11y=20
5x-11y=59
20.x-4y=11
8x-6y-36

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

1.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax-1%2Ay=1
6%2Ax-3%2Ay=3

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-1%2Ay=1-2%2AxSubtract 2%2Ax from both sides

y=%281-2%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-1%2B2%2Ax Now we've fully isolated y

Since y equals -1%2B2%2Ax we can substitute the expression -1%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


6%2Ax%2B-3%2Ahighlight%28%28-1%2B2%2Ax%29%29=3 Replace y with -1%2B2%2Ax. Since this eliminates y, we can now solve for x.

6%2Ax-3%2A%28-1%29-3%282%29x=3 Distribute -3 to -1%2B2%2Ax

6%2Ax%2B3-6%2Ax=3 Multiply



6%2Ax%2B3-6%2Ax=3 Reduce any fractions

6%2Ax-6%2Ax=3-3 Subtract 3 from both sides


6%2Ax-6%2Ax=0 Combine the terms on the right side



0%2Ax=0 Now combine the terms on the left side.
0=0 Since this expression is true for any x, we have an identity.


So there are an infinite number solutions. The simple reason is the 2 equations represent 2 lines that overlap each other. So they intersect each other at an infinite number of points.

If we graph 2%2Ax-1%2Ay=1 and 6%2Ax-3%2Ay=3 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%281-2%2Ax%29%2F-1%29+ graph of 2%2Ax-1%2Ay=1


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%283-6%2Ax%29%2F-3+%29+ graph of 6%2Ax-3%2Ay=3 (hint: you may have to solve for y to graph these)

we can see that these two lines are the same. So this system is dependent



2.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-5%2Ax%2B1%2Ay=-1
-5%2Ax%2B1%2Ay=4

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=-1%2B5%2AxAdd 5%2Ax to both sides

y=%28-1%2B5%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-1%2B5%2Ax Now we've fully isolated y

Since y equals -1%2B5%2Ax we can substitute the expression -1%2B5%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-5%2Ax%2B1%2Ahighlight%28%28-1%2B5%2Ax%29%29=4 Replace y with -1%2B5%2Ax. Since this eliminates y, we can now solve for x.

-5%2Ax%2B1%2A%28-1%29%2B1%285%29x=4 Distribute 1 to -1%2B5%2Ax

-5%2Ax-1%2B5%2Ax=4 Multiply



-5%2Ax-1%2B5%2Ax=4 Reduce any fractions

-5%2Ax%2B5%2Ax=4%2B1Add 1 to both sides


-5%2Ax%2B5%2Ax=5 Combine the terms on the right side



0%2Ax=5 Now combine the terms on the left side.
0%2F1=5%2F1 Since this expression is not true, we have an inconsistency.


So there are no solutions. The simple reason is the 2 equations represent 2 parallel lines that will never intersect. Since no intersections occur, no solutions exist.


graph of -5%2Ax%2B1%2Ay=-1 (red) and -5%2Ax%2B1%2Ay=4 (green) (hint: you may have to solve for y to graph these)


and we can see that the two equations are parallel and will never intersect. So this system is inconsistent


3.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=38
1%2Ax-2%2Ay=-25

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=38-1%2AxSubtract 1%2Ax from both sides

y=%2838-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=38-1%2Ax Now we've fully isolated y

Since y equals 38-1%2Ax we can substitute the expression 38-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-2%2Ahighlight%28%2838-1%2Ax%29%29=-25 Replace y with 38-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-2%2A%2838%29-2%28-1%29x=-25 Distribute -2 to 38-1%2Ax

1%2Ax-76%2B2%2Ax=-25 Multiply



1%2Ax-76%2B2%2Ax=-25 Reduce any fractions

1%2Ax%2B2%2Ax=-25%2B76Add 76 to both sides


1%2Ax%2B2%2Ax=51 Combine the terms on the right side



3%2Ax=51 Now combine the terms on the left side.


cross%28%281%2F3%29%283%2F1%29%29x=%2851%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3%2F1 and isolate x

So when we multiply 51%2F1 and 1%2F3 (and simplify) we get



x=17 <---------------------------------One answer

Now that we know that x=17, lets substitute that in for x to solve for y

1%2817%29-2%2Ay=-25 Plug in x=17 into the 2nd equation

17-2%2Ay=-25 Multiply

-2%2Ay=-25-17Subtract 17 from both sides

-2%2Ay=-42 Combine the terms on the right side

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-42%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=-42%2F-2 Multiply the terms on the right side


y=21 Reduce


So this is the other answer


y=21<---------------------------------Other answer


So our solution is

x=17 and y=21

which can also look like

(17,21)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=38
1%2Ax-2%2Ay=-25

we get


graph of 1%2Ax%2B1%2Ay=38 (red) and 1%2Ax-2%2Ay=-25 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (17,21). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (17,21) into the system of equations


Let x=17 and y=21. Now plug those values into the equation 1%2Ax%2B1%2Ay=38

1%2A%2817%29%2B1%2A%2821%29=38 Plug in x=17 and y=21


17%2B21=38 Multiply


38=38 Add


38=38 Reduce. Since this equation is true the solution works.


So the solution (17,21) satisfies 1%2Ax%2B1%2Ay=38



Let x=17 and y=21. Now plug those values into the equation 1%2Ax-2%2Ay=-25

1%2A%2817%29-2%2A%2821%29=-25 Plug in x=17 and y=21


17-42=-25 Multiply


-25=-25 Add


-25=-25 Reduce. Since this equation is true the solution works.


So the solution (17,21) satisfies 1%2Ax-2%2Ay=-25


Since the solution (17,21) satisfies the system of equations


1%2Ax%2B1%2Ay=38
1%2Ax-2%2Ay=-25


this verifies our answer.





4.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-2%2Ax%2B5%2Ay=9
1%2Ax%2B1%2Ay=13

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

5%2Ay=9%2B2%2AxAdd 2%2Ax to both sides

y=%289%2B2%2Ax%29%2F5 Divide both sides by 5.


Which breaks down and reduces to



y=9%2F5%2B%282%2F5%29%2Ax Now we've fully isolated y

Since y equals 9%2F5%2B%282%2F5%29%2Ax we can substitute the expression 9%2F5%2B%282%2F5%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%289%2F5%2B%282%2F5%29%2Ax%29%29=13 Replace y with 9%2F5%2B%282%2F5%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%289%2F5%29%2B1%282%2F5%29x=13 Distribute 1 to 9%2F5%2B%282%2F5%29%2Ax

1%2Ax%2B9%2F5%2B%282%2F5%29%2Ax=13 Multiply



1%2Ax%2B9%2F5%2B%282%2F5%29%2Ax=13 Reduce any fractions

1%2Ax%2B%282%2F5%29%2Ax=13-9%2F5 Subtract 9%2F5 from both sides


1%2Ax%2B%282%2F5%29%2Ax=65%2F5-9%2F5 Make 13 into a fraction with a denominator of 5


1%2Ax%2B%282%2F5%29%2Ax=56%2F5 Combine the terms on the right side



%285%2F5%29%2Ax%2B%282%2F5%29x=56%2F5 Make 1 into a fraction with a denominator of 5

%287%2F5%29%2Ax=56%2F5 Now combine the terms on the left side.


cross%28%285%2F7%29%287%2F5%29%29x=%2856%2F5%29%285%2F7%29 Multiply both sides by 5%2F7. This will cancel out 7%2F5 and isolate x

So when we multiply 56%2F5 and 5%2F7 (and simplify) we get



x=8 <---------------------------------One answer

Now that we know that x=8, lets substitute that in for x to solve for y

1%288%29%2B1%2Ay=13 Plug in x=8 into the 2nd equation

8%2B1%2Ay=13 Multiply

1%2Ay=13-8Subtract 8 from both sides

1%2Ay=5 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%285%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=5%2F1 Multiply the terms on the right side


y=5 Reduce


So this is the other answer


y=5<---------------------------------Other answer


So our solution is

x=8 and y=5

which can also look like

(8,5)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-2%2Ax%2B5%2Ay=9
1%2Ax%2B1%2Ay=13

we get


graph of -2%2Ax%2B5%2Ay=9 (red) and 1%2Ax%2B1%2Ay=13 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (8,5). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (8,5) into the system of equations


Let x=8 and y=5. Now plug those values into the equation -2%2Ax%2B5%2Ay=9

-2%2A%288%29%2B5%2A%285%29=9 Plug in x=8 and y=5


-16%2B25=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (8,5) satisfies -2%2Ax%2B5%2Ay=9



Let x=8 and y=5. Now plug those values into the equation 1%2Ax%2B1%2Ay=13

1%2A%288%29%2B1%2A%285%29=13 Plug in x=8 and y=5


8%2B5=13 Multiply


13=13 Add


13=13 Reduce. Since this equation is true the solution works.


So the solution (8,5) satisfies 1%2Ax%2B1%2Ay=13


Since the solution (8,5) satisfies the system of equations


-2%2Ax%2B5%2Ay=9
1%2Ax%2B1%2Ay=13


this verifies our answer.




5.

Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

2%2Ax%2B1%2Ay=7
3%2Ax-1%2Ay=3

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=7-2%2AxSubtract 2%2Ax from both sides

y=%287-2%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=7-2%2Ax Now we've fully isolated y

Since y equals 7-2%2Ax we can substitute the expression 7-2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B-1%2Ahighlight%28%287-2%2Ax%29%29=3 Replace y with 7-2%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax-1%2A%287%29-1%28-2%29x=3 Distribute -1 to 7-2%2Ax

3%2Ax-7%2B2%2Ax=3 Multiply



3%2Ax-7%2B2%2Ax=3 Reduce any fractions

3%2Ax%2B2%2Ax=3%2B7Add 7 to both sides


3%2Ax%2B2%2Ax=10 Combine the terms on the right side



5%2Ax=10 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%2810%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply 10%2F1 and 1%2F5 (and simplify) we get



x=2 <---------------------------------One answer

Now that we know that x=2, lets substitute that in for x to solve for y

3%282%29-1%2Ay=3 Plug in x=2 into the 2nd equation

6-1%2Ay=3 Multiply

-1%2Ay=3-6Subtract 6 from both sides

-1%2Ay=-3 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%28-3%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=-3%2F-1 Multiply the terms on the right side


y=3 Reduce


So this is the other answer


y=3<---------------------------------Other answer


So our solution is

x=2 and y=3

which can also look like

(2,3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

2%2Ax%2B1%2Ay=7
3%2Ax-1%2Ay=3

we get


graph of 2%2Ax%2B1%2Ay=7 (red) and 3%2Ax-1%2Ay=3 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (2,3). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (2,3) into the system of equations


Let x=2 and y=3. Now plug those values into the equation 2%2Ax%2B1%2Ay=7

2%2A%282%29%2B1%2A%283%29=7 Plug in x=2 and y=3


4%2B3=7 Multiply


7=7 Add


7=7 Reduce. Since this equation is true the solution works.


So the solution (2,3) satisfies 2%2Ax%2B1%2Ay=7



Let x=2 and y=3. Now plug those values into the equation 3%2Ax-1%2Ay=3

3%2A%282%29-1%2A%283%29=3 Plug in x=2 and y=3


6-3=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


So the solution (2,3) satisfies 3%2Ax-1%2Ay=3


Since the solution (2,3) satisfies the system of equations


2%2Ax%2B1%2Ay=7
3%2Ax-1%2Ay=3


this verifies our answer.




6.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-3%2Ax%2B1%2Ay=2
2%2Ax%2B1%2Ay=-8

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=2%2B3%2AxAdd 3%2Ax to both sides

y=%282%2B3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=2%2B3%2Ax Now we've fully isolated y

Since y equals 2%2B3%2Ax we can substitute the expression 2%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%282%2B3%2Ax%29%29=-8 Replace y with 2%2B3%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%282%29%2B1%283%29x=-8 Distribute 1 to 2%2B3%2Ax

2%2Ax%2B2%2B3%2Ax=-8 Multiply



2%2Ax%2B2%2B3%2Ax=-8 Reduce any fractions

2%2Ax%2B3%2Ax=-8-2 Subtract 2 from both sides


2%2Ax%2B3%2Ax=-10 Combine the terms on the right side



5%2Ax=-10 Now combine the terms on the left side.


cross%28%281%2F5%29%285%2F1%29%29x=%28-10%2F1%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5%2F1 and isolate x

So when we multiply -10%2F1 and 1%2F5 (and simplify) we get



x=-2 <---------------------------------One answer

Now that we know that x=-2, lets substitute that in for x to solve for y

2%28-2%29%2B1%2Ay=-8 Plug in x=-2 into the 2nd equation

-4%2B1%2Ay=-8 Multiply

1%2Ay=-8%2B4Add 4 to both sides

1%2Ay=-4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%28-4%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=-4%2F1 Multiply the terms on the right side


y=-4 Reduce


So this is the other answer


y=-4<---------------------------------Other answer


So our solution is

x=-2 and y=-4

which can also look like

(-2,-4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-3%2Ax%2B1%2Ay=2
2%2Ax%2B1%2Ay=-8

we get


graph of -3%2Ax%2B1%2Ay=2 (red) and 2%2Ax%2B1%2Ay=-8 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-2,-4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-2,-4) into the system of equations


Let x=-2 and y=-4. Now plug those values into the equation -3%2Ax%2B1%2Ay=2

-3%2A%28-2%29%2B1%2A%28-4%29=2 Plug in x=-2 and y=-4


6-4=2 Multiply


2=2 Add


2=2 Reduce. Since this equation is true the solution works.


So the solution (-2,-4) satisfies -3%2Ax%2B1%2Ay=2



Let x=-2 and y=-4. Now plug those values into the equation 2%2Ax%2B1%2Ay=-8

2%2A%28-2%29%2B1%2A%28-4%29=-8 Plug in x=-2 and y=-4


-4-4=-8 Multiply


-8=-8 Add


-8=-8 Reduce. Since this equation is true the solution works.


So the solution (-2,-4) satisfies 2%2Ax%2B1%2Ay=-8


Since the solution (-2,-4) satisfies the system of equations


-3%2Ax%2B1%2Ay=2
2%2Ax%2B1%2Ay=-8


this verifies our answer.





7.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=-3
2%2Ax%2B1%2Ay=-8

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=-3-1%2AxSubtract 1%2Ax from both sides

y=%28-3-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-3-1%2Ax Now we've fully isolated y

Since y equals -3-1%2Ax we can substitute the expression -3-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%28-3-1%2Ax%29%29=-8 Replace y with -3-1%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%28-3%29%2B1%28-1%29x=-8 Distribute 1 to -3-1%2Ax

2%2Ax-3-1%2Ax=-8 Multiply



2%2Ax-3-1%2Ax=-8 Reduce any fractions

2%2Ax-1%2Ax=-8%2B3Add 3 to both sides


2%2Ax-1%2Ax=-5 Combine the terms on the right side



1%2Ax=-5 Now combine the terms on the left side.


cross%28%281%2F1%29%281%2F1%29%29x=%28-5%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1%2F1 and isolate x

So when we multiply -5%2F1 and 1%2F1 (and simplify) we get



x=-5 <---------------------------------One answer

Now that we know that x=-5, lets substitute that in for x to solve for y

2%28-5%29%2B1%2Ay=-8 Plug in x=-5 into the 2nd equation

-10%2B1%2Ay=-8 Multiply

1%2Ay=-8%2B10Add 10 to both sides

1%2Ay=2 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%282%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=2%2F1 Multiply the terms on the right side


y=2 Reduce


So this is the other answer


y=2<---------------------------------Other answer


So our solution is

x=-5 and y=2

which can also look like

(-5,2)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=-3
2%2Ax%2B1%2Ay=-8

we get


graph of 1%2Ax%2B1%2Ay=-3 (red) and 2%2Ax%2B1%2Ay=-8 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-5,2). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-5,2) into the system of equations


Let x=-5 and y=2. Now plug those values into the equation 1%2Ax%2B1%2Ay=-3

1%2A%28-5%29%2B1%2A%282%29=-3 Plug in x=-5 and y=2


-5%2B2=-3 Multiply


-3=-3 Add


-3=-3 Reduce. Since this equation is true the solution works.


So the solution (-5,2) satisfies 1%2Ax%2B1%2Ay=-3



Let x=-5 and y=2. Now plug those values into the equation 2%2Ax%2B1%2Ay=-8

2%2A%28-5%29%2B1%2A%282%29=-8 Plug in x=-5 and y=2


-10%2B2=-8 Multiply


-8=-8 Add


-8=-8 Reduce. Since this equation is true the solution works.


So the solution (-5,2) satisfies 2%2Ax%2B1%2Ay=-8


Since the solution (-5,2) satisfies the system of equations


1%2Ax%2B1%2Ay=-3
2%2Ax%2B1%2Ay=-8


this verifies our answer.




8. y%3C_3x-2 y%3Ex%2B1} shade area where these statements are true
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax%2B1%2Ay=-2
-1%2Ax%2B1%2Ay=1

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=-2-3%2AxSubtract 3%2Ax from both sides

y=%28-2-3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-2-3%2Ax Now we've fully isolated y

Since y equals -2-3%2Ax we can substitute the expression -2-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B1%2Ahighlight%28%28-2-3%2Ax%29%29=1 Replace y with -2-3%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B1%2A%28-2%29%2B1%28-3%29x=1 Distribute 1 to -2-3%2Ax

-1%2Ax-2-3%2Ax=1 Multiply



-1%2Ax-2-3%2Ax=1 Reduce any fractions

-1%2Ax-3%2Ax=1%2B2Add 2 to both sides


-1%2Ax-3%2Ax=3 Combine the terms on the right side



-4%2Ax=3 Now combine the terms on the left side.


cross%28%281%2F-4%29%28-4%2F1%29%29x=%283%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4%2F1 and isolate x

So when we multiply 3%2F1 and 1%2F-4 (and simplify) we get



x=-3%2F4 <---------------------------------One answer

Now that we know that x=-3%2F4, lets substitute that in for x to solve for y

-1%28-3%2F4%29%2B1%2Ay=1 Plug in x=-3%2F4 into the 2nd equation

3%2F4%2B1%2Ay=1 Multiply

1%2Ay=1-3%2F4Subtract 3%2F4 from both sides

1%2Ay=4%2F4-3%2F4 Make 1 into a fraction with a denominator of 4



1%2Ay=1%2F4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%281%2F4%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=1%2F4 Multiply the terms on the right side


y=1%2F4 Reduce


So this is the other answer


y=1%2F4<---------------------------------Other answer


So our solution is

x=-3%2F4 and y=1%2F4

which can also look like

(-3%2F4,1%2F4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax%2B1%2Ay=-2
-1%2Ax%2B1%2Ay=1

we get


graph of 3%2Ax%2B1%2Ay=-2 (red) and -1%2Ax%2B1%2Ay=1 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-3%2F4,1%2F4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-3%2F4,1%2F4) into the system of equations


Let x=-3%2F4 and y=1%2F4. Now plug those values into the equation 3%2Ax%2B1%2Ay=-2

3%2A%28-3%2F4%29%2B1%2A%281%2F4%29=-2 Plug in x=-3%2F4 and y=1%2F4


-9%2F4%2B1%2F4=-2 Multiply


-8%2F4=-2 Add


-2=-2 Reduce. Since this equation is true the solution works.


So the solution (-3%2F4,1%2F4) satisfies 3%2Ax%2B1%2Ay=-2



Let x=-3%2F4 and y=1%2F4. Now plug those values into the equation -1%2Ax%2B1%2Ay=1

-1%2A%28-3%2F4%29%2B1%2A%281%2F4%29=1 Plug in x=-3%2F4 and y=1%2F4


3%2F4%2B1%2F4=1 Multiply


4%2F4=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (-3%2F4,1%2F4) satisfies -1%2Ax%2B1%2Ay=1


Since the solution (-3%2F4,1%2F4) satisfies the system of equations


3%2Ax%2B1%2Ay=-2
-1%2Ax%2B1%2Ay=1


this verifies our answer.




9.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-1%2Ax%2B1%2Ay=0
-2%2Ax%2B1%2Ay=-4

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=0%2B1%2AxAdd 1%2Ax to both sides

y=%280%2B1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=0%2B1%2Ax Now we've fully isolated y

Since y equals 0%2B1%2Ax we can substitute the expression 0%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-2%2Ax%2B1%2Ahighlight%28%280%2B1%2Ax%29%29=-4 Replace y with 0%2B1%2Ax. Since this eliminates y, we can now solve for x.

-2%2Ax%2B1%2A%280%29%2B1%281%29x=-4 Distribute 1 to 0%2B1%2Ax

-2%2Ax%2B0%2B1%2Ax=-4 Multiply



-2%2Ax%2B0%2B1%2Ax=-4 Reduce any fractions

-2%2Ax%2B1%2Ax=-4%2B0Add 0 to both sides


-2%2Ax%2B1%2Ax=-4 Combine the terms on the right side



-1%2Ax=-4 Now combine the terms on the left side.


cross%28%281%2F-1%29%28-1%2F1%29%29x=%28-4%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1%2F1 and isolate x

So when we multiply -4%2F1 and 1%2F-1 (and simplify) we get



x=4 <---------------------------------One answer

Now that we know that x=4, lets substitute that in for x to solve for y

-2%284%29%2B1%2Ay=-4 Plug in x=4 into the 2nd equation

-8%2B1%2Ay=-4 Multiply

1%2Ay=-4%2B8Add 8 to both sides

1%2Ay=4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%284%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=4%2F1 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=4 and y=4

which can also look like

(4,4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax%2B1%2Ay=0
-2%2Ax%2B1%2Ay=-4

we get


graph of -1%2Ax%2B1%2Ay=0 (red) and -2%2Ax%2B1%2Ay=-4 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (4,4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (4,4) into the system of equations


Let x=4 and y=4. Now plug those values into the equation -1%2Ax%2B1%2Ay=0

-1%2A%284%29%2B1%2A%284%29=0 Plug in x=4 and y=4


-4%2B4=0 Multiply


0=0 Add


0=0 Reduce. Since this equation is true the solution works.


So the solution (4,4) satisfies -1%2Ax%2B1%2Ay=0



Let x=4 and y=4. Now plug those values into the equation -2%2Ax%2B1%2Ay=-4

-2%2A%284%29%2B1%2A%284%29=-4 Plug in x=4 and y=4


-8%2B4=-4 Multiply


-4=-4 Add


-4=-4 Reduce. Since this equation is true the solution works.


So the solution (4,4) satisfies -2%2Ax%2B1%2Ay=-4


Since the solution (4,4) satisfies the system of equations


-1%2Ax%2B1%2Ay=0
-2%2Ax%2B1%2Ay=-4


this verifies our answer.




10.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-2%2Ax%2B1%2Ay=6
-3%2Ax%2B1%2Ay=9

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=6%2B2%2AxAdd 2%2Ax to both sides

y=%286%2B2%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=6%2B2%2Ax Now we've fully isolated y

Since y equals 6%2B2%2Ax we can substitute the expression 6%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-3%2Ax%2B1%2Ahighlight%28%286%2B2%2Ax%29%29=9 Replace y with 6%2B2%2Ax. Since this eliminates y, we can now solve for x.

-3%2Ax%2B1%2A%286%29%2B1%282%29x=9 Distribute 1 to 6%2B2%2Ax

-3%2Ax%2B6%2B2%2Ax=9 Multiply



-3%2Ax%2B6%2B2%2Ax=9 Reduce any fractions

-3%2Ax%2B2%2Ax=9-6 Subtract 6 from both sides


-3%2Ax%2B2%2Ax=3 Combine the terms on the right side



-1%2Ax=3 Now combine the terms on the left side.


cross%28%281%2F-1%29%28-1%2F1%29%29x=%283%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1%2F1 and isolate x

So when we multiply 3%2F1 and 1%2F-1 (and simplify) we get



x=-3 <---------------------------------One answer

Now that we know that x=-3, lets substitute that in for x to solve for y

-3%28-3%29%2B1%2Ay=9 Plug in x=-3 into the 2nd equation

9%2B1%2Ay=9 Multiply

1%2Ay=9-9Subtract 9 from both sides

1%2Ay=0 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%280%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=0%2F1 Multiply the terms on the right side


y=0 Reduce


So this is the other answer


y=0<---------------------------------Other answer


So our solution is

x=-3 and y=0

which can also look like

(-3,0)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-2%2Ax%2B1%2Ay=6
-3%2Ax%2B1%2Ay=9

we get


graph of -2%2Ax%2B1%2Ay=6 (red) and -3%2Ax%2B1%2Ay=9 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-3,0). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-3,0) into the system of equations


Let x=-3 and y=0. Now plug those values into the equation -2%2Ax%2B1%2Ay=6

-2%2A%28-3%29%2B1%2A%280%29=6 Plug in x=-3 and y=0


6%2B0=6 Multiply


6=6 Add


6=6 Reduce. Since this equation is true the solution works.


So the solution (-3,0) satisfies -2%2Ax%2B1%2Ay=6



Let x=-3 and y=0. Now plug those values into the equation -3%2Ax%2B1%2Ay=9

-3%2A%28-3%29%2B1%2A%280%29=9 Plug in x=-3 and y=0


9%2B0=9 Multiply


9=9 Add


9=9 Reduce. Since this equation is true the solution works.


So the solution (-3,0) satisfies -3%2Ax%2B1%2Ay=9


Since the solution (-3,0) satisfies the system of equations


-2%2Ax%2B1%2Ay=6
-3%2Ax%2B1%2Ay=9


this verifies our answer.




11. y%3C3x-2
-3x%2By-2 here is something missing
12.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=8
1%2Ax-1%2Ay=10

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=8-1%2AxSubtract 1%2Ax from both sides

y=%288-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=8-1%2Ax Now we've fully isolated y

Since y equals 8-1%2Ax we can substitute the expression 8-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%288-1%2Ax%29%29=10 Replace y with 8-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%288%29-1%28-1%29x=10 Distribute -1 to 8-1%2Ax

1%2Ax-8%2B1%2Ax=10 Multiply



1%2Ax-8%2B1%2Ax=10 Reduce any fractions

1%2Ax%2B1%2Ax=10%2B8Add 8 to both sides


1%2Ax%2B1%2Ax=18 Combine the terms on the right side



2%2Ax=18 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2818%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 18%2F1 and 1%2F2 (and simplify) we get



x=9 <---------------------------------One answer

Now that we know that x=9, lets substitute that in for x to solve for y

1%289%29-1%2Ay=10 Plug in x=9 into the 2nd equation

9-1%2Ay=10 Multiply

-1%2Ay=10-9Subtract 9 from both sides

-1%2Ay=1 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%281%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=1%2F-1 Multiply the terms on the right side


y=-1 Reduce


So this is the other answer


y=-1<---------------------------------Other answer


So our solution is

x=9 and y=-1

which can also look like

(9,-1)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=8
1%2Ax-1%2Ay=10

we get


graph of 1%2Ax%2B1%2Ay=8 (red) and 1%2Ax-1%2Ay=10 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (9,-1). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (9,-1) into the system of equations


Let x=9 and y=-1. Now plug those values into the equation 1%2Ax%2B1%2Ay=8

1%2A%289%29%2B1%2A%28-1%29=8 Plug in x=9 and y=-1


9-1=8 Multiply


8=8 Add


8=8 Reduce. Since this equation is true the solution works.


So the solution (9,-1) satisfies 1%2Ax%2B1%2Ay=8



Let x=9 and y=-1. Now plug those values into the equation 1%2Ax-1%2Ay=10

1%2A%289%29-1%2A%28-1%29=10 Plug in x=9 and y=-1


9%2B1=10 Multiply


10=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (9,-1) satisfies 1%2Ax-1%2Ay=10


Since the solution (9,-1) satisfies the system of equations


1%2Ax%2B1%2Ay=8
1%2Ax-1%2Ay=10


this verifies our answer.




13.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-1%2Ax%2B1%2Ay=-3
3%2Ax%2B1%2Ay=25

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=-3%2B1%2AxAdd 1%2Ax to both sides

y=%28-3%2B1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-3%2B1%2Ax Now we've fully isolated y

Since y equals -3%2B1%2Ax we can substitute the expression -3%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B1%2Ahighlight%28%28-3%2B1%2Ax%29%29=25 Replace y with -3%2B1%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B1%2A%28-3%29%2B1%281%29x=25 Distribute 1 to -3%2B1%2Ax

3%2Ax-3%2B1%2Ax=25 Multiply



3%2Ax-3%2B1%2Ax=25 Reduce any fractions

3%2Ax%2B1%2Ax=25%2B3Add 3 to both sides


3%2Ax%2B1%2Ax=28 Combine the terms on the right side



4%2Ax=28 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%2828%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply 28%2F1 and 1%2F4 (and simplify) we get



x=7 <---------------------------------One answer

Now that we know that x=7, lets substitute that in for x to solve for y

3%287%29%2B1%2Ay=25 Plug in x=7 into the 2nd equation

21%2B1%2Ay=25 Multiply

1%2Ay=25-21Subtract 21 from both sides

1%2Ay=4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%284%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=4%2F1 Multiply the terms on the right side


y=4 Reduce


So this is the other answer


y=4<---------------------------------Other answer


So our solution is

x=7 and y=4

which can also look like

(7,4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax%2B1%2Ay=-3
3%2Ax%2B1%2Ay=25

we get


graph of -1%2Ax%2B1%2Ay=-3 (red) and 3%2Ax%2B1%2Ay=25 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (7,4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (7,4) into the system of equations


Let x=7 and y=4. Now plug those values into the equation -1%2Ax%2B1%2Ay=-3

-1%2A%287%29%2B1%2A%284%29=-3 Plug in x=7 and y=4


-7%2B4=-3 Multiply


-3=-3 Add


-3=-3 Reduce. Since this equation is true the solution works.


So the solution (7,4) satisfies -1%2Ax%2B1%2Ay=-3



Let x=7 and y=4. Now plug those values into the equation 3%2Ax%2B1%2Ay=25

3%2A%287%29%2B1%2A%284%29=25 Plug in x=7 and y=4


21%2B4=25 Multiply


25=25 Add


25=25 Reduce. Since this equation is true the solution works.


So the solution (7,4) satisfies 3%2Ax%2B1%2Ay=25


Since the solution (7,4) satisfies the system of equations


-1%2Ax%2B1%2Ay=-3
3%2Ax%2B1%2Ay=25


this verifies our answer.




14.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax-2%2Ay=1
2%2Ax%2B1%2Ay=4

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-2%2Ay=1-4%2AxSubtract 4%2Ax from both sides

y=%281-4%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-1%2F2%2B2%2Ax Now we've fully isolated y

Since y equals -1%2F2%2B2%2Ax we can substitute the expression -1%2F2%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%28-1%2F2%2B2%2Ax%29%29=4 Replace y with -1%2F2%2B2%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%28-1%2F2%29%2B1%282%29x=4 Distribute 1 to -1%2F2%2B2%2Ax

2%2Ax-1%2F2%2B2%2Ax=4 Multiply



2%2Ax-1%2F2%2B2%2Ax=4 Reduce any fractions

2%2Ax%2B2%2Ax=4%2B1%2F2Add 1%2F2 to both sides


2%2Ax%2B2%2Ax=8%2F2%2B1%2F2 Make 4 into a fraction with a denominator of 2


2%2Ax%2B2%2Ax=9%2F2 Combine the terms on the right side



4%2Ax=9%2F2 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%289%2F2%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply 9%2F2 and 1%2F4 (and simplify) we get



x=9%2F8 <---------------------------------One answer

Now that we know that x=9%2F8, lets substitute that in for x to solve for y

2%289%2F8%29%2B1%2Ay=4 Plug in x=9%2F8 into the 2nd equation

9%2F4%2B1%2Ay=4 Multiply

1%2Ay=4-9%2F4Subtract 9%2F4 from both sides

1%2Ay=16%2F4-9%2F4 Make 4 into a fraction with a denominator of 4



1%2Ay=7%2F4 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%287%2F4%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=7%2F4 Multiply the terms on the right side


y=7%2F4 Reduce


So this is the other answer


y=7%2F4<---------------------------------Other answer


So our solution is

x=9%2F8 and y=7%2F4

which can also look like

(9%2F8,7%2F4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

4%2Ax-2%2Ay=1
2%2Ax%2B1%2Ay=4

we get


graph of 4%2Ax-2%2Ay=1 (red) and 2%2Ax%2B1%2Ay=4 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (9%2F8,7%2F4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (9%2F8,7%2F4) into the system of equations


Let x=9%2F8 and y=7%2F4. Now plug those values into the equation 4%2Ax-2%2Ay=1

4%2A%289%2F8%29-2%2A%287%2F4%29=1 Plug in x=9%2F8 and y=7%2F4


36%2F8-14%2F4=1 Multiply


8%2F8=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (9%2F8,7%2F4) satisfies 4%2Ax-2%2Ay=1



Let x=9%2F8 and y=7%2F4. Now plug those values into the equation 2%2Ax%2B1%2Ay=4

2%2A%289%2F8%29%2B1%2A%287%2F4%29=4 Plug in x=9%2F8 and y=7%2F4


18%2F8%2B7%2F4=4 Multiply


32%2F8=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (9%2F8,7%2F4) satisfies 2%2Ax%2B1%2Ay=4


Since the solution (9%2F8,7%2F4) satisfies the system of equations


4%2Ax-2%2Ay=1
2%2Ax%2B1%2Ay=4


this verifies our answer.





15.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax-2%2Ay=10
2%2Ax%2B4%2Ay=10

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-2%2Ay=10-1%2AxSubtract 1%2Ax from both sides

y=%2810-1%2Ax%29%2F-2 Divide both sides by -2.


Which breaks down and reduces to



y=-5%2B%281%2F2%29%2Ax Now we've fully isolated y

Since y equals -5%2B%281%2F2%29%2Ax we can substitute the expression -5%2B%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B4%2Ahighlight%28%28-5%2B%281%2F2%29%2Ax%29%29=10 Replace y with -5%2B%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B4%2A%28-5%29%2B4%281%2F2%29x=10 Distribute 4 to -5%2B%281%2F2%29%2Ax

2%2Ax-20%2B%284%2F2%29%2Ax=10 Multiply



2%2Ax-20%2B2%2Ax=10 Reduce any fractions

2%2Ax%2B2%2Ax=10%2B20Add 20 to both sides


2%2Ax%2B2%2Ax=30 Combine the terms on the right side



4%2Ax=30 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%2830%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply 30%2F1 and 1%2F4 (and simplify) we get



x=15%2F2 <---------------------------------One answer

Now that we know that x=15%2F2, lets substitute that in for x to solve for y

2%2815%2F2%29%2B4%2Ay=10 Plug in x=15%2F2 into the 2nd equation

15%2B4%2Ay=10 Multiply

4%2Ay=10-15Subtract 15 from both sides

4%2Ay=-5 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ay=%28-5%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.

y=-5%2F4 Multiply the terms on the right side


y=-5%2F4 Reduce


So this is the other answer


y=-5%2F4<---------------------------------Other answer


So our solution is

x=15%2F2 and y=-5%2F4

which can also look like

(15%2F2,-5%2F4)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax-2%2Ay=10
2%2Ax%2B4%2Ay=10

we get


graph of 1%2Ax-2%2Ay=10 (red) and 2%2Ax%2B4%2Ay=10 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (15%2F2,-5%2F4). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (15%2F2,-5%2F4) into the system of equations


Let x=15%2F2 and y=-5%2F4. Now plug those values into the equation 1%2Ax-2%2Ay=10

1%2A%2815%2F2%29-2%2A%28-5%2F4%29=10 Plug in x=15%2F2 and y=-5%2F4


15%2F2%2B10%2F4=10 Multiply


40%2F4=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (15%2F2,-5%2F4) satisfies 1%2Ax-2%2Ay=10



Let x=15%2F2 and y=-5%2F4. Now plug those values into the equation 2%2Ax%2B4%2Ay=10

2%2A%2815%2F2%29%2B4%2A%28-5%2F4%29=10 Plug in x=15%2F2 and y=-5%2F4


30%2F2-20%2F4=10 Multiply


40%2F4=10 Add


10=10 Reduce. Since this equation is true the solution works.


So the solution (15%2F2,-5%2F4) satisfies 2%2Ax%2B4%2Ay=10


Since the solution (15%2F2,-5%2F4) satisfies the system of equations


1%2Ax-2%2Ay=10
2%2Ax%2B4%2Ay=10


this verifies our answer.




16.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-1%2Ax%2B1%2Ay=4
-4%2Ax%2B1%2Ay=1

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=4%2B1%2AxAdd 1%2Ax to both sides

y=%284%2B1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=4%2B1%2Ax Now we've fully isolated y

Since y equals 4%2B1%2Ax we can substitute the expression 4%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-4%2Ax%2B1%2Ahighlight%28%284%2B1%2Ax%29%29=1 Replace y with 4%2B1%2Ax. Since this eliminates y, we can now solve for x.

-4%2Ax%2B1%2A%284%29%2B1%281%29x=1 Distribute 1 to 4%2B1%2Ax

-4%2Ax%2B4%2B1%2Ax=1 Multiply



-4%2Ax%2B4%2B1%2Ax=1 Reduce any fractions

-4%2Ax%2B1%2Ax=1-4 Subtract 4 from both sides


-4%2Ax%2B1%2Ax=-3 Combine the terms on the right side



-3%2Ax=-3 Now combine the terms on the left side.


cross%28%281%2F-3%29%28-3%2F1%29%29x=%28-3%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3%2F1 and isolate x

So when we multiply -3%2F1 and 1%2F-3 (and simplify) we get



x=1 <---------------------------------One answer

Now that we know that x=1, lets substitute that in for x to solve for y

-4%281%29%2B1%2Ay=1 Plug in x=1 into the 2nd equation

-4%2B1%2Ay=1 Multiply

1%2Ay=1%2B4Add 4 to both sides

1%2Ay=5 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%285%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=5%2F1 Multiply the terms on the right side


y=5 Reduce


So this is the other answer


y=5<---------------------------------Other answer


So our solution is

x=1 and y=5

which can also look like

(1,5)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-1%2Ax%2B1%2Ay=4
-4%2Ax%2B1%2Ay=1

we get


graph of -1%2Ax%2B1%2Ay=4 (red) and -4%2Ax%2B1%2Ay=1 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (1,5). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (1,5) into the system of equations


Let x=1 and y=5. Now plug those values into the equation -1%2Ax%2B1%2Ay=4

-1%2A%281%29%2B1%2A%285%29=4 Plug in x=1 and y=5


-1%2B5=4 Multiply


4=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (1,5) satisfies -1%2Ax%2B1%2Ay=4



Let x=1 and y=5. Now plug those values into the equation -4%2Ax%2B1%2Ay=1

-4%2A%281%29%2B1%2A%285%29=1 Plug in x=1 and y=5


-4%2B5=1 Multiply


1=1 Add


1=1 Reduce. Since this equation is true the solution works.


So the solution (1,5) satisfies -4%2Ax%2B1%2Ay=1


Since the solution (1,5) satisfies the system of equations


-1%2Ax%2B1%2Ay=4
-4%2Ax%2B1%2Ay=1


this verifies our answer.




17.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

-6%2Ax%2B1%2Ay=-4
2%2Ax%2B1%2Ay=28

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=-4%2B6%2AxAdd 6%2Ax to both sides

y=%28-4%2B6%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-4%2B6%2Ax Now we've fully isolated y

Since y equals -4%2B6%2Ax we can substitute the expression -4%2B6%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%28-4%2B6%2Ax%29%29=28 Replace y with -4%2B6%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%28-4%29%2B1%286%29x=28 Distribute 1 to -4%2B6%2Ax

2%2Ax-4%2B6%2Ax=28 Multiply



2%2Ax-4%2B6%2Ax=28 Reduce any fractions

2%2Ax%2B6%2Ax=28%2B4Add 4 to both sides


2%2Ax%2B6%2Ax=32 Combine the terms on the right side



8%2Ax=32 Now combine the terms on the left side.


cross%28%281%2F8%29%288%2F1%29%29x=%2832%2F1%29%281%2F8%29 Multiply both sides by 1%2F8. This will cancel out 8%2F1 and isolate x

So when we multiply 32%2F1 and 1%2F8 (and simplify) we get



x=4 <---------------------------------One answer

Now that we know that x=4, lets substitute that in for x to solve for y

2%284%29%2B1%2Ay=28 Plug in x=4 into the 2nd equation

8%2B1%2Ay=28 Multiply

1%2Ay=28-8Subtract 8 from both sides

1%2Ay=20 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%2820%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=20%2F1 Multiply the terms on the right side


y=20 Reduce


So this is the other answer


y=20<---------------------------------Other answer


So our solution is

x=4 and y=20

which can also look like

(4,20)

Notice if we graph the equations (if you need help with graphing, check out this solver)

-6%2Ax%2B1%2Ay=-4
2%2Ax%2B1%2Ay=28

we get


graph of -6%2Ax%2B1%2Ay=-4 (red) and 2%2Ax%2B1%2Ay=28 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (4,20). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (4,20) into the system of equations


Let x=4 and y=20. Now plug those values into the equation -6%2Ax%2B1%2Ay=-4

-6%2A%284%29%2B1%2A%2820%29=-4 Plug in x=4 and y=20


-24%2B20=-4 Multiply


-4=-4 Add


-4=-4 Reduce. Since this equation is true the solution works.


So the solution (4,20) satisfies -6%2Ax%2B1%2Ay=-4



Let x=4 and y=20. Now plug those values into the equation 2%2Ax%2B1%2Ay=28

2%2A%284%29%2B1%2A%2820%29=28 Plug in x=4 and y=20


8%2B20=28 Multiply


28=28 Add


28=28 Reduce. Since this equation is true the solution works.


So the solution (4,20) satisfies 2%2Ax%2B1%2Ay=28


Since the solution (4,20) satisfies the system of equations


-6%2Ax%2B1%2Ay=-4
2%2Ax%2B1%2Ay=28


this verifies our answer.




18.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax-10%2Ay=-25
4%2Ax-40%2Ay=20

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-10%2Ay=-25-3%2AxSubtract 3%2Ax from both sides

y=%28-25-3%2Ax%29%2F-10 Divide both sides by -10.


Which breaks down and reduces to



y=5%2F2%2B%283%2F10%29%2Ax Now we've fully isolated y

Since y equals 5%2F2%2B%283%2F10%29%2Ax we can substitute the expression 5%2F2%2B%283%2F10%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-40%2Ahighlight%28%285%2F2%2B%283%2F10%29%2Ax%29%29=20 Replace y with 5%2F2%2B%283%2F10%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-40%2A%285%2F2%29-40%283%2F10%29x=20 Distribute -40 to 5%2F2%2B%283%2F10%29%2Ax

4%2Ax-200%2F2-%28120%2F10%29%2Ax=20 Multiply



4%2Ax-100-12%2Ax=20 Reduce any fractions

4%2Ax-12%2Ax=20%2B100Add 100 to both sides


4%2Ax-12%2Ax=120 Combine the terms on the right side



-8%2Ax=120 Now combine the terms on the left side.


cross%28%281%2F-8%29%28-8%2F1%29%29x=%28120%2F1%29%281%2F-8%29 Multiply both sides by 1%2F-8. This will cancel out -8%2F1 and isolate x

So when we multiply 120%2F1 and 1%2F-8 (and simplify) we get



x=-15 <---------------------------------One answer

Now that we know that x=-15, lets substitute that in for x to solve for y

4%28-15%29-40%2Ay=20 Plug in x=-15 into the 2nd equation

-60-40%2Ay=20 Multiply

-40%2Ay=20%2B60Add 60 to both sides

-40%2Ay=80 Combine the terms on the right side

cross%28%281%2F-40%29%28-40%29%29%2Ay=%2880%2F1%29%281%2F-40%29 Multiply both sides by 1%2F-40. This will cancel out -40 on the left side.

y=80%2F-40 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=-15 and y=-2

which can also look like

(-15,-2)

Notice if we graph the equations (if you need help with graphing, check out this solver)

3%2Ax-10%2Ay=-25
4%2Ax-40%2Ay=20

we get


graph of 3%2Ax-10%2Ay=-25 (red) and 4%2Ax-40%2Ay=20 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-15,-2). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-15,-2) into the system of equations


Let x=-15 and y=-2. Now plug those values into the equation 3%2Ax-10%2Ay=-25

3%2A%28-15%29-10%2A%28-2%29=-25 Plug in x=-15 and y=-2


-45%2B20=-25 Multiply


-25=-25 Add


-25=-25 Reduce. Since this equation is true the solution works.


So the solution (-15,-2) satisfies 3%2Ax-10%2Ay=-25



Let x=-15 and y=-2. Now plug those values into the equation 4%2Ax-40%2Ay=20

4%2A%28-15%29-40%2A%28-2%29=20 Plug in x=-15 and y=-2


-60%2B80=20 Multiply


20=20 Add


20=20 Reduce. Since this equation is true the solution works.


So the solution (-15,-2) satisfies 4%2Ax-40%2Ay=20


Since the solution (-15,-2) satisfies the system of equations


3%2Ax-10%2Ay=-25
4%2Ax-40%2Ay=20


this verifies our answer.




19.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

8%2Ax-11%2Ay=20
5%2Ax-11%2Ay=59

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-11%2Ay=20-8%2AxSubtract 8%2Ax from both sides

y=%2820-8%2Ax%29%2F-11 Divide both sides by -11.


Which breaks down and reduces to



y=-20%2F11%2B%288%2F11%29%2Ax Now we've fully isolated y

Since y equals -20%2F11%2B%288%2F11%29%2Ax we can substitute the expression -20%2F11%2B%288%2F11%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


5%2Ax%2B-11%2Ahighlight%28%28-20%2F11%2B%288%2F11%29%2Ax%29%29=59 Replace y with -20%2F11%2B%288%2F11%29%2Ax. Since this eliminates y, we can now solve for x.

5%2Ax-11%2A%28-20%2F11%29-11%288%2F11%29x=59 Distribute -11 to -20%2F11%2B%288%2F11%29%2Ax

5%2Ax%2B220%2F11-%2888%2F11%29%2Ax=59 Multiply



5%2Ax%2B20-8%2Ax=59 Reduce any fractions

5%2Ax-8%2Ax=59-20 Subtract 20 from both sides


5%2Ax-8%2Ax=39 Combine the terms on the right side



-3%2Ax=39 Now combine the terms on the left side.


cross%28%281%2F-3%29%28-3%2F1%29%29x=%2839%2F1%29%281%2F-3%29 Multiply both sides by 1%2F-3. This will cancel out -3%2F1 and isolate x

So when we multiply 39%2F1 and 1%2F-3 (and simplify) we get



x=-13 <---------------------------------One answer

Now that we know that x=-13, lets substitute that in for x to solve for y

5%28-13%29-11%2Ay=59 Plug in x=-13 into the 2nd equation

-65-11%2Ay=59 Multiply

-11%2Ay=59%2B65Add 65 to both sides

-11%2Ay=124 Combine the terms on the right side

cross%28%281%2F-11%29%28-11%29%29%2Ay=%28124%2F1%29%281%2F-11%29 Multiply both sides by 1%2F-11. This will cancel out -11 on the left side.

y=124%2F-11 Multiply the terms on the right side


y=-124%2F11 Reduce


So this is the other answer


y=-124%2F11<---------------------------------Other answer


So our solution is

x=-13 and y=-124%2F11

which can also look like

(-13,-124%2F11)

Notice if we graph the equations (if you need help with graphing, check out this solver)

8%2Ax-11%2Ay=20
5%2Ax-11%2Ay=59

we get


graph of 8%2Ax-11%2Ay=20 (red) and 5%2Ax-11%2Ay=59 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (-13,-124%2F11). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (-13,-124%2F11) into the system of equations


Let x=-13 and y=-124%2F11. Now plug those values into the equation 8%2Ax-11%2Ay=20

8%2A%28-13%29-11%2A%28-124%2F11%29=20 Plug in x=-13 and y=-124%2F11


-104%2B1364%2F11=20 Multiply


220%2F11=20 Add


20=20 Reduce. Since this equation is true the solution works.


So the solution (-13,-124%2F11) satisfies 8%2Ax-11%2Ay=20



Let x=-13 and y=-124%2F11. Now plug those values into the equation 5%2Ax-11%2Ay=59

5%2A%28-13%29-11%2A%28-124%2F11%29=59 Plug in x=-13 and y=-124%2F11


-65%2B1364%2F11=59 Multiply


649%2F11=59 Add


59=59 Reduce. Since this equation is true the solution works.


So the solution (-13,-124%2F11) satisfies 5%2Ax-11%2Ay=59


Since the solution (-13,-124%2F11) satisfies the system of equations


8%2Ax-11%2Ay=20
5%2Ax-11%2Ay=59


this verifies our answer.





20.
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax-4%2Ay=11
8%2Ax-6%2Ay=36

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

-4%2Ay=11-1%2AxSubtract 1%2Ax from both sides

y=%2811-1%2Ax%29%2F-4 Divide both sides by -4.


Which breaks down and reduces to



y=-11%2F4%2B%281%2F4%29%2Ax Now we've fully isolated y

Since y equals -11%2F4%2B%281%2F4%29%2Ax we can substitute the expression -11%2F4%2B%281%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


8%2Ax%2B-6%2Ahighlight%28%28-11%2F4%2B%281%2F4%29%2Ax%29%29=36 Replace y with -11%2F4%2B%281%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

8%2Ax-6%2A%28-11%2F4%29-6%281%2F4%29x=36 Distribute -6 to -11%2F4%2B%281%2F4%29%2Ax

8%2Ax%2B66%2F4-%286%2F4%29%2Ax=36 Multiply



8%2Ax%2B33%2F2-%283%2F2%29%2Ax=36 Reduce any fractions

8%2Ax-%283%2F2%29%2Ax=36-33%2F2 Subtract 33%2F2 from both sides


8%2Ax-%283%2F2%29%2Ax=72%2F2-33%2F2 Make 36 into a fraction with a denominator of 2


8%2Ax-%283%2F2%29%2Ax=39%2F2 Combine the terms on the right side



%2816%2F2%29%2Ax-%283%2F2%29x=39%2F2 Make 8 into a fraction with a denominator of 2

%2813%2F2%29%2Ax=39%2F2 Now combine the terms on the left side.


cross%28%282%2F13%29%2813%2F2%29%29x=%2839%2F2%29%282%2F13%29 Multiply both sides by 2%2F13. This will cancel out 13%2F2 and isolate x

So when we multiply 39%2F2 and 2%2F13 (and simplify) we get



x=3 <---------------------------------One answer

Now that we know that x=3, lets substitute that in for x to solve for y

8%283%29-6%2Ay=36 Plug in x=3 into the 2nd equation

24-6%2Ay=36 Multiply

-6%2Ay=36-24Subtract 24 from both sides

-6%2Ay=12 Combine the terms on the right side

cross%28%281%2F-6%29%28-6%29%29%2Ay=%2812%2F1%29%281%2F-6%29 Multiply both sides by 1%2F-6. This will cancel out -6 on the left side.

y=12%2F-6 Multiply the terms on the right side


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=3 and y=-2

which can also look like

(3,-2)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax-4%2Ay=11
8%2Ax-6%2Ay=36

we get


graph of 1%2Ax-4%2Ay=11 (red) and 8%2Ax-6%2Ay=36 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (3,-2). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (3,-2) into the system of equations


Let x=3 and y=-2. Now plug those values into the equation 1%2Ax-4%2Ay=11

1%2A%283%29-4%2A%28-2%29=11 Plug in x=3 and y=-2


3%2B8=11 Multiply


11=11 Add


11=11 Reduce. Since this equation is true the solution works.


So the solution (3,-2) satisfies 1%2Ax-4%2Ay=11



Let x=3 and y=-2. Now plug those values into the equation 8%2Ax-6%2Ay=36

8%2A%283%29-6%2A%28-2%29=36 Plug in x=3 and y=-2


24%2B12=36 Multiply


36=36 Add


36=36 Reduce. Since this equation is true the solution works.


So the solution (3,-2) satisfies 8%2Ax-6%2Ay=36


Since the solution (3,-2) satisfies the system of equations


1%2Ax-4%2Ay=11
8%2Ax-6%2Ay=36


this verifies our answer.