SOLUTION: Solve each system using substitution.write no solution or infinitely many solutions where appropriate.(1)x=-2y+1. x=y-5 (2) y=x-7.2x+y=8 (3)3x+y=10.y=-3x+4 (4)4x+2y=8.y=-2x+4 (5)5x

Algebra ->  Linear-equations -> SOLUTION: Solve each system using substitution.write no solution or infinitely many solutions where appropriate.(1)x=-2y+1. x=y-5 (2) y=x-7.2x+y=8 (3)3x+y=10.y=-3x+4 (4)4x+2y=8.y=-2x+4 (5)5x      Log On


   

Question 1010610: Solve each system using substitution.write no solution or infinitely many solutions where appropriate.(1)x=-2y+1. x=y-5 (2) y=x-7.2x+y=8 (3)3x+y=10.y=-3x+4 (4)4x+2y=8.y=-2x+4 (5)5x-3y=-4.x+y=-4 how to solve this?
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

(1)
x=-2y%2B1
x=y-5+
--------------------
x%2B2y=1
x-y=-5+
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B2%2Ay=1
1%2Ax-1%2Ay=-5

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

y=%281-1%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



y=1%2F2-%281%2F2%29%2Ax Now we've fully isolated y

Since y equals 1%2F2-%281%2F2%29%2Ax we can substitute the expression 1%2F2-%281%2F2%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%281%2F2-%281%2F2%29%2Ax%29%29=-5 Replace y with 1%2F2-%281%2F2%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%281%2F2%29-1%28-1%2F2%29x=-5 Distribute -1 to 1%2F2-%281%2F2%29%2Ax

1%2Ax-1%2F2%2B%281%2F2%29%2Ax=-5 Multiply



1%2Ax-1%2F2%2B%281%2F2%29%2Ax=-5 Reduce any fractions

1%2Ax%2B%281%2F2%29%2Ax=-5%2B1%2F2Add 1%2F2 to both sides


1%2Ax%2B%281%2F2%29%2Ax=-10%2F2%2B1%2F2 Make -5 into a fraction with a denominator of 2


1%2Ax%2B%281%2F2%29%2Ax=-9%2F2 Combine the terms on the right side



%282%2F2%29%2Ax%2B%281%2F2%29x=-9%2F2 Make 1 into a fraction with a denominator of 2

%283%2F2%29%2Ax=-9%2F2 Now combine the terms on the left side.


cross%28%282%2F3%29%283%2F2%29%29x=%28-9%2F2%29%282%2F3%29 Multiply both sides by 2%2F3. This will cancel out 3%2F2 and isolate x

So when we multiply -9%2F2 and 2%2F3 (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

1%28-3%29-1%2Ay=-5 Plug in x=-3 into the 2nd equation

-3-1%2Ay=-5 Multiply

-1%2Ay=-5%2B3Add 3 to both sides

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

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

y=-2%2F-1 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%2B2%2Ay=1
1%2Ax-1%2Ay=-5

we get


graph of 1%2Ax%2B2%2Ay=1 (red) and 1%2Ax-1%2Ay=-5 (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%2B2%2Ay=1

1%2A%28-3%29%2B2%2A%282%29=1 Plug in x=-3 and y=2


-3%2B4=1 Multiply


1=1 Add


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


So the solution (-3,2) satisfies 1%2Ax%2B2%2Ay=1



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

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


-3-2=-5 Multiply


-5=-5 Add


-5=-5 Reduce. Since this equation is true the solution works.


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


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


1%2Ax%2B2%2Ay=1
1%2Ax-1%2Ay=-5


this verifies our answer.



(2)
y=x-7
2x%2By=8
-------------------
-x%2By=-7
2x%2By=8
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=-7
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=-7%2B1%2AxAdd 1%2Ax to both sides

y=%28-7%2B1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-7%2B1%2Ax Now we've fully isolated y

Since y equals -7%2B1%2Ax we can substitute the expression -7%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B1%2Ahighlight%28%28-7%2B1%2Ax%29%29=8 Replace y with -7%2B1%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax%2B1%2A%28-7%29%2B1%281%29x=8 Distribute 1 to -7%2B1%2Ax

2%2Ax-7%2B1%2Ax=8 Multiply



2%2Ax-7%2B1%2Ax=8 Reduce any fractions

2%2Ax%2B1%2Ax=8%2B7Add 7 to both sides


2%2Ax%2B1%2Ax=15 Combine the terms on the right side



3%2Ax=15 Now combine the terms on the left side.


cross%28%281%2F3%29%283%2F1%29%29x=%2815%2F1%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3%2F1 and isolate x

So when we multiply 15%2F1 and 1%2F3 (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%285%29%2B1%2Ay=8 Plug in x=5 into the 2nd equation

10%2B1%2Ay=8 Multiply

1%2Ay=8-10Subtract 10 from both sides

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

cross%28%281%2F1%29%281%29%29%2Ay=%28-2%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=-7
2%2Ax%2B1%2Ay=8

we get


graph of -1%2Ax%2B1%2Ay=-7 (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=-7

-1%2A%285%29%2B1%2A%28-2%29=-7 Plug in x=5 and y=-2


-5-2=-7 Multiply


-7=-7 Add


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


So the solution (5,-2) satisfies -1%2Ax%2B1%2Ay=-7



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

2%2A%285%29%2B1%2A%28-2%29=8 Plug in x=5 and y=-2


10-2=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=-7
2%2Ax%2B1%2Ay=8


this verifies our answer.





(3)
3x%2By=10
y=-3x%2B4
-------------------------
3x%2By=10
3x%2By=4
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=10
3%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=10-3%2AxSubtract 3%2Ax from both sides

y=%2810-3%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=10-3%2Ax Now we've fully isolated y

Since y equals 10-3%2Ax we can substitute the expression 10-3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


3%2Ax%2B1%2Ahighlight%28%2810-3%2Ax%29%29=4 Replace y with 10-3%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B1%2A%2810%29%2B1%28-3%29x=4 Distribute 1 to 10-3%2Ax

3%2Ax%2B10-3%2Ax=4 Multiply



3%2Ax%2B10-3%2Ax=4 Reduce any fractions

3%2Ax-3%2Ax=4-10 Subtract 10 from both sides


3%2Ax-3%2Ax=-6 Combine the terms on the right side



0%2Ax=-6 Now combine the terms on the left side.
0%2F1=-6%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%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%2810-3%2Ax%29%2F1%2C+%284-3%2Ax%29%2F1+%29+ graph of 3%2Ax%2B1%2Ay=10 (red) and 3%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


(4)
4x%2B2y=8
y=-2x%2B4
-------------------------
4x%2B2y=8
2x%2By=4
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

4%2Ax%2B2%2Ay=8
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=8-4%2AxSubtract 4%2Ax from both sides

y=%288-4%2Ax%29%2F2 Divide both sides by 2.


Which breaks down and reduces to



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

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


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

2%2Ax%2B1%2A%284%29%2B1%28-2%29x=4 Distribute 1 to 4-2%2Ax

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



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

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


2%2Ax-2%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 4%2Ax%2B2%2Ay=8 and 2%2Ax%2B1%2Ay=4 we get

+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%288-4%2Ax%29%2F2%29+ graph of 4%2Ax%2B2%2Ay=8


+graph%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%284-2%2Ax%29%2F1+%29+ graph of 2%2Ax%2B1%2Ay=4 (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


(5)
5x-3y=-4
x%2By=-4+
---------------------------
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

5%2Ax-3%2Ay=-4
1%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

-3%2Ay=-4-5%2AxSubtract 5%2Ax from both sides

y=%28-4-5%2Ax%29%2F-3 Divide both sides by -3.


Which breaks down and reduces to



y=4%2F3%2B%285%2F3%29%2Ax Now we've fully isolated y

Since y equals 4%2F3%2B%285%2F3%29%2Ax we can substitute the expression 4%2F3%2B%285%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%284%2F3%2B%285%2F3%29%2Ax%29%29=-4 Replace y with 4%2F3%2B%285%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%284%2F3%29%2B1%285%2F3%29x=-4 Distribute 1 to 4%2F3%2B%285%2F3%29%2Ax

1%2Ax%2B4%2F3%2B%285%2F3%29%2Ax=-4 Multiply



1%2Ax%2B4%2F3%2B%285%2F3%29%2Ax=-4 Reduce any fractions

1%2Ax%2B%285%2F3%29%2Ax=-4-4%2F3 Subtract 4%2F3 from both sides


1%2Ax%2B%285%2F3%29%2Ax=-12%2F3-4%2F3 Make -4 into a fraction with a denominator of 3


1%2Ax%2B%285%2F3%29%2Ax=-16%2F3 Combine the terms on the right side



%283%2F3%29%2Ax%2B%285%2F3%29x=-16%2F3 Make 1 into a fraction with a denominator of 3

%288%2F3%29%2Ax=-16%2F3 Now combine the terms on the left side.


cross%28%283%2F8%29%288%2F3%29%29x=%28-16%2F3%29%283%2F8%29 Multiply both sides by 3%2F8. This will cancel out 8%2F3 and isolate x

So when we multiply -16%2F3 and 3%2F8 (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

1%28-2%29%2B1%2Ay=-4 Plug in x=-2 into the 2nd equation

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

1%2Ay=-4%2B2Add 2 to both sides

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

cross%28%281%2F1%29%281%29%29%2Ay=%28-2%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=-2 and y=-2

which can also look like

(-2,-2)

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

5%2Ax-3%2Ay=-4
1%2Ax%2B1%2Ay=-4

we get


graph of 5%2Ax-3%2Ay=-4 (red) and 1%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 (-2,-2). This verifies our answer.


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

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


Let x=-2 and y=-2. Now plug those values into the equation 5%2Ax-3%2Ay=-4

5%2A%28-2%29-3%2A%28-2%29=-4 Plug in x=-2 and y=-2


-10%2B6=-4 Multiply


-4=-4 Add


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


So the solution (-2,-2) satisfies 5%2Ax-3%2Ay=-4



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

1%2A%28-2%29%2B1%2A%28-2%29=-4 Plug in x=-2 and y=-2


-2-2=-4 Multiply


-4=-4 Add


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


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


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


5%2Ax-3%2Ay=-4
1%2Ax%2B1%2Ay=-4


this verifies our answer.