SOLUTION: use the method of substitution to solve the system of linear equations 2x-y=8 9x-4y=37

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Question 688008: use the method of substitution to solve the system of linear equations
2x-y=8
9x-4y=37
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

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=8
9%2Ax-4%2Ay=37

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

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


Which breaks down and reduces to



y=-8%2B2%2Ax Now we've fully isolated y

Since y equals -8%2B2%2Ax we can substitute the expression -8%2B2%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


9%2Ax%2B-4%2Ahighlight%28%28-8%2B2%2Ax%29%29=37 Replace y with -8%2B2%2Ax. Since this eliminates y, we can now solve for x.

9%2Ax-4%2A%28-8%29-4%282%29x=37 Distribute -4 to -8%2B2%2Ax

9%2Ax%2B32-8%2Ax=37 Multiply



9%2Ax%2B32-8%2Ax=37 Reduce any fractions

9%2Ax-8%2Ax=37-32 Subtract 32 from both sides


9%2Ax-8%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

9%285%29-4%2Ay=37 Plug in x=5 into the 2nd equation

45-4%2Ay=37 Multiply

-4%2Ay=37-45Subtract 45 from both sides

-4%2Ay=-8 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ay=%28-8%2F1%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 on the left side.

y=-8%2F-4 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)

2%2Ax-1%2Ay=8
9%2Ax-4%2Ay=37

we get


graph of 2%2Ax-1%2Ay=8 (red) and 9%2Ax-4%2Ay=37 (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.


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Check:

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


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

2%2A%285%29-1%2A%282%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-1%2Ay=8



Let x=5 and y=2. Now plug those values into the equation 9%2Ax-4%2Ay=37

9%2A%285%29-4%2A%282%29=37 Plug in x=5 and y=2


45-8=37 Multiply


37=37 Add


37=37 Reduce. Since this equation is true the solution works.


So the solution (5,2) satisfies 9%2Ax-4%2Ay=37


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


2%2Ax-1%2Ay=8
9%2Ax-4%2Ay=37


this verifies our answer.