SOLUTION: What are the steps for solving this problem: Solve the linear system of equations- 3x + 7y = 15 -5x + 2y = 16

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Question 118797: What are the steps for solving this problem:
Solve the linear system of equations-
3x + 7y = 15
-5x + 2y = 16
Answer by jim_thompson5910(35256) 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

3%2Ax%2B7%2Ay=15
-5%2Ax%2B2%2Ay=16

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

7%2Ay=15-3%2AxSubtract 3%2Ax from both sides

y=%2815-3%2Ax%29%2F7 Divide both sides by 7.


Which breaks down and reduces to



y=15%2F7-%283%2F7%29%2Ax Now we've fully isolated y

Since y equals 15%2F7-%283%2F7%29%2Ax we can substitute the expression 15%2F7-%283%2F7%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-5%2Ax%2B2%2Ahighlight%28%2815%2F7-%283%2F7%29%2Ax%29%29=16 Replace y with 15%2F7-%283%2F7%29%2Ax. Since this eliminates y, we can now solve for x.

-5%2Ax%2B2%2A%2815%2F7%29%2B2%28-3%2F7%29x=16 Distribute 2 to 15%2F7-%283%2F7%29%2Ax

-5%2Ax%2B30%2F7-%286%2F7%29%2Ax=16 Multiply



-5%2Ax%2B30%2F7-%286%2F7%29%2Ax=16 Reduce any fractions

-5%2Ax-%286%2F7%29%2Ax=16-30%2F7 Subtract 30%2F7 from both sides


-5%2Ax-%286%2F7%29%2Ax=112%2F7-30%2F7 Make 16 into a fraction with a denominator of 7


-5%2Ax-%286%2F7%29%2Ax=82%2F7 Combine the terms on the right side



%28-35%2F7%29%2Ax-%286%2F7%29x=82%2F7 Make -5 into a fraction with a denominator of 7

%28-41%2F7%29%2Ax=82%2F7 Now combine the terms on the left side.


cross%28%287%2F-41%29%28-41%2F7%29%29x=%2882%2F7%29%287%2F-41%29 Multiply both sides by 7%2F-41. This will cancel out -41%2F7 and isolate x

So when we multiply 82%2F7 and 7%2F-41 (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

-5%28-2%29%2B2%2Ay=16 Plug in x=-2 into the 2nd equation

10%2B2%2Ay=16 Multiply

2%2Ay=16-10Subtract 10 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ay=%286%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.

y=6%2F2 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)

3%2Ax%2B7%2Ay=15
-5%2Ax%2B2%2Ay=16

we get


graph of 3%2Ax%2B7%2Ay=15 (red) and -5%2Ax%2B2%2Ay=16 (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.


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Check:

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


Let x=-2 and y=3. Now plug those values into the equation 3%2Ax%2B7%2Ay=15

3%2A%28-2%29%2B7%2A%283%29=15 Plug in x=-2 and y=3


-6%2B21=15 Multiply


15=15 Add


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


So the solution (-2,3) satisfies 3%2Ax%2B7%2Ay=15



Let x=-2 and y=3. Now plug those values into the equation -5%2Ax%2B2%2Ay=16

-5%2A%28-2%29%2B2%2A%283%29=16 Plug in x=-2 and y=3


10%2B6=16 Multiply


16=16 Add


16=16 Reduce. Since this equation is true the solution works.


So the solution (-2,3) satisfies -5%2Ax%2B2%2Ay=16


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


3%2Ax%2B7%2Ay=15
-5%2Ax%2B2%2Ay=16


this verifies our answer.