SOLUTION: Solve the system by substitution. 3x – y = –7 x + y = –9

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Question 117214: Solve the system by substitution.
3x – y = –7
x + y = –9
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-1%2Ay=-7
1%2Ax%2B1%2Ay=-9

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

y=%28-7-3%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=7%2B3%2Ax Now we've fully isolated y

Since y equals 7%2B3%2Ax we can substitute the expression 7%2B3%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%287%2B3%2Ax%29%29=-9 Replace y with 7%2B3%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%287%29%2B1%283%29x=-9 Distribute 1 to 7%2B3%2Ax

1%2Ax%2B7%2B3%2Ax=-9 Multiply



1%2Ax%2B7%2B3%2Ax=-9 Reduce any fractions

1%2Ax%2B3%2Ax=-9-7 Subtract 7 from both sides


1%2Ax%2B3%2Ax=-16 Combine the terms on the right side



4%2Ax=-16 Now combine the terms on the left side.


cross%28%281%2F4%29%284%2F1%29%29x=%28-16%2F1%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4%2F1 and isolate x

So when we multiply -16%2F1 and 1%2F4 (and simplify) we get



x=-4 <---------------------------------One answer

Now that we know that x=-4, lets substitute that in for x to solve for y

1%28-4%29%2B1%2Ay=-9 Plug in x=-4 into the 2nd equation

-4%2B1%2Ay=-9 Multiply

1%2Ay=-9%2B4Add 4 to both sides

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

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

y=-5%2F1 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=-4 and y=-5

which can also look like

(-4,-5)

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

3%2Ax-1%2Ay=-7
1%2Ax%2B1%2Ay=-9

we get


graph of 3%2Ax-1%2Ay=-7 (red) and 1%2Ax%2B1%2Ay=-9 (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 (-4,-5). This verifies our answer.


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Check:

Plug in (-4,-5) into the system of equations


Let x=-4 and y=-5. Now plug those values into the equation 3%2Ax-1%2Ay=-7

3%2A%28-4%29-1%2A%28-5%29=-7 Plug in x=-4 and y=-5


-12%2B5=-7 Multiply


-7=-7 Add


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


So the solution (-4,-5) satisfies 3%2Ax-1%2Ay=-7



Let x=-4 and y=-5. Now plug those values into the equation 1%2Ax%2B1%2Ay=-9

1%2A%28-4%29%2B1%2A%28-5%29=-9 Plug in x=-4 and y=-5


-4-5=-9 Multiply


-9=-9 Add


-9=-9 Reduce. Since this equation is true the solution works.


So the solution (-4,-5) satisfies 1%2Ax%2B1%2Ay=-9


Since the solution (-4,-5) satisfies the system of equations


3%2Ax-1%2Ay=-7
1%2Ax%2B1%2Ay=-9


this verifies our answer.