SOLUTION: The solution to the system of equations x+3y=12 4x-y=-17 is

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Question 90137This question is from textbook
: The solution to the system of equations x+3y=12 4x-y=-17 is This question is from textbook

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

1%2Ax%2B3%2Ay=12
4%2Ax-1%2Ay=-17

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=12-1%2AxSubtract 1%2Ax from both sides

y=%2812-1%2Ax%29%2F3 Divide both sides by 3.


Which breaks down and reduces to



y=4-%281%2F3%29%2Ax Now we've fully isolated y

Since y equals 4-%281%2F3%29%2Ax we can substitute the expression 4-%281%2F3%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


4%2Ax%2B-1%2Ahighlight%28%284-%281%2F3%29%2Ax%29%29=-17 Replace y with 4-%281%2F3%29%2Ax. Since this eliminates y, we can now solve for x.

4%2Ax-1%2A%284%29-1%28-1%2F3%29x=-17 Distribute -1 to 4-%281%2F3%29%2Ax

4%2Ax-4%2B%281%2F3%29%2Ax=-17 Multiply



4%2Ax-4%2B%281%2F3%29%2Ax=-17 Reduce any fractions

4%2Ax%2B%281%2F3%29%2Ax=-17%2B4Add 4 to both sides


4%2Ax%2B%281%2F3%29%2Ax=-13 Combine the terms on the right side



%2812%2F3%29%2Ax%2B%281%2F3%29x=-13 Make 4 into a fraction with a denominator of 3

%2813%2F3%29%2Ax=-13 Now combine the terms on the left side.


cross%28%283%2F13%29%2813%2F3%29%29x=%28-13%2F1%29%283%2F13%29 Multiply both sides by 3%2F13. This will cancel out 13%2F3 and isolate x

So when we multiply -13%2F1 and 3%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

4%28-3%29-1%2Ay=-17 Plug in x=-3 into the 2nd equation

-12-1%2Ay=-17 Multiply

-1%2Ay=-17%2B12Add 12 to both sides

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

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

y=-5%2F-1 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=-3 and y=5

which can also look like

(-3,5)

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

1%2Ax%2B3%2Ay=12
4%2Ax-1%2Ay=-17

we get


graph of 1%2Ax%2B3%2Ay=12 (red) and 4%2Ax-1%2Ay=-17 (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 (-3,5). This verifies our answer.


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Check:

Plug in (-3,5) into the system of equations


Let x=-3 and y=5. Now plug those values into the equation 1%2Ax%2B3%2Ay=12

1%2A%28-3%29%2B3%2A%285%29=12 Plug in x=-3 and y=5


-3%2B15=12 Multiply


12=12 Add


12=12 Reduce. Since this equation is true the solution works.


So the solution (-3,5) satisfies 1%2Ax%2B3%2Ay=12



Let x=-3 and y=5. Now plug those values into the equation 4%2Ax-1%2Ay=-17

4%2A%28-3%29-1%2A%285%29=-17 Plug in x=-3 and y=5


-12-5=-17 Multiply


-17=-17 Add


-17=-17 Reduce. Since this equation is true the solution works.


So the solution (-3,5) satisfies 4%2Ax-1%2Ay=-17


Since the solution (-3,5) satisfies the system of equations


1%2Ax%2B3%2Ay=12
4%2Ax-1%2Ay=-17


this verifies our answer.