SOLUTION: 1) Use the method of substitution to solve the system of linear equations 3x-y=5 7x-2y=15 2) use the method of substitution to solve lineear equations 3a=2-b 3a=7-b 3

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: 1) Use the method of substitution to solve the system of linear equations 3x-y=5 7x-2y=15 2) use the method of substitution to solve lineear equations 3a=2-b 3a=7-b 3      Log On


   

Question 672717: 1) Use the method of substitution to solve the system of linear equations
3x-y=5
7x-2y=15
2) use the method of substitution to solve lineear equations
3a=2-b
3a=7-b
3)Solve the system of linear equations
3x-y=0
2x-y=1
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

1.
3x-y=5
7x-2y=15
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax-1%2Ay=5
7%2Ax-2%2Ay=15

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

y=%285-3%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-5%2B3%2Ax Now we've fully isolated y

Since y equals -5%2B3%2Ax we can substitute the expression -5%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


7%2Ax%2B-2%2Ahighlight%28%28-5%2B3%2Ax%29%29=15 Replace y with -5%2B3%2Ax. Since this eliminates y, we can now solve for x.

7%2Ax-2%2A%28-5%29-2%283%29x=15 Distribute -2 to -5%2B3%2Ax

7%2Ax%2B10-6%2Ax=15 Multiply



7%2Ax%2B10-6%2Ax=15 Reduce any fractions

7%2Ax-6%2Ax=15-10 Subtract 10 from both sides


7%2Ax-6%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=%285%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

7%285%29-2%2Ay=15 Plug in x=5 into the 2nd equation

35-2%2Ay=15 Multiply

-2%2Ay=15-35Subtract 35 from both sides

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

cross%28%281%2F-2%29%28-2%29%29%2Ay=%28-20%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2 on the left side.

y=-20%2F-2 Multiply the terms on the right side


y=10 Reduce


So this is the other answer


y=10<---------------------------------Other answer


So our solution is

x=5 and y=10

which can also look like

(5,10)

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

3%2Ax-1%2Ay=5
7%2Ax-2%2Ay=15

we get


graph of 3%2Ax-1%2Ay=5 (red) and 7%2Ax-2%2Ay=15 (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 (5,10). This verifies our answer.


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Check:

Plug in (5,10) into the system of equations


Let x=5 and y=10. Now plug those values into the equation 3%2Ax-1%2Ay=5

3%2A%285%29-1%2A%2810%29=5 Plug in x=5 and y=10


15-10=5 Multiply


5=5 Add


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


So the solution (5,10) satisfies 3%2Ax-1%2Ay=5



Let x=5 and y=10. Now plug those values into the equation 7%2Ax-2%2Ay=15

7%2A%285%29-2%2A%2810%29=15 Plug in x=5 and y=10


35-20=15 Multiply


15=15 Add


15=15 Reduce. Since this equation is true the solution works.


So the solution (5,10) satisfies 7%2Ax-2%2Ay=15


Since the solution (5,10) satisfies the system of equations


3%2Ax-1%2Ay=5
7%2Ax-2%2Ay=15


this verifies our answer.





2.
3a=2-b
3a=7-b
---------------
3a%2Bb=2......let x=a and y=b
3a%2Bb=7
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
3%2Ax%2B1%2Ay=7

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=%282-3%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.


3%2Ax%2B1%2Ahighlight%28%282-3%2Ax%29%29=7 Replace y with 2-3%2Ax. Since this eliminates y, we can now solve for x.

3%2Ax%2B1%2A%282%29%2B1%28-3%29x=7 Distribute 1 to 2-3%2Ax

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



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

3%2Ax-3%2Ax=7-2 Subtract 2 from both sides


3%2Ax-3%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%28+500%2C+600%2C+-6%2C+5%2C+-10%2C+10%2C+%282-3%2Ax%29%2F1%2C+%287-3%2Ax%29%2F1+%29+ graph of 3%2Ax%2B1%2Ay=2 (red) and 3%2Ax%2B1%2Ay=7 (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.
3x-y=0
2x-y=1
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

3%2Ax-1%2Ay=0
2%2Ax-1%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=0-3%2AxSubtract 3%2Ax from both sides

y=%280-3%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=0%2B3%2Ax Now we've fully isolated y

Since y equals 0%2B3%2Ax we can substitute the expression 0%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


2%2Ax%2B-1%2Ahighlight%28%280%2B3%2Ax%29%29=1 Replace y with 0%2B3%2Ax. Since this eliminates y, we can now solve for x.

2%2Ax-1%2A%280%29-1%283%29x=1 Distribute -1 to 0%2B3%2Ax

2%2Ax%2B0-3%2Ax=1 Multiply



2%2Ax%2B0-3%2Ax=1 Reduce any fractions

2%2Ax-3%2Ax=1%2B0Add 0 to both sides


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



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


cross%28%281%2F-1%29%28-1%2F1%29%29x=%281%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 1%2F1 and 1%2F-1 (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

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

-2-1%2Ay=1 Multiply

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

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

cross%28%281%2F-1%29%28-1%29%29%2Ay=%283%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=-1 and y=-3

which can also look like

(-1,-3)

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

3%2Ax-1%2Ay=0
2%2Ax-1%2Ay=1

we get


graph of 3%2Ax-1%2Ay=0 (red) and 2%2Ax-1%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,-3). This verifies our answer.


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

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


Let x=-1 and y=-3. Now plug those values into the equation 3%2Ax-1%2Ay=0

3%2A%28-1%29-1%2A%28-3%29=0 Plug in x=-1 and y=-3


-3%2B3=0 Multiply


0=0 Add


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


So the solution (-1,-3) satisfies 3%2Ax-1%2Ay=0



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

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


-2%2B3=1 Multiply


1=1 Add


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


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


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


3%2Ax-1%2Ay=0
2%2Ax-1%2Ay=1


this verifies our answer.