SOLUTION: Show all your work to receive full credit for this problem. Use the Substitution method to solve the system of equations. Give your answer as an ordered pair (x,y). x +

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Question 88442: Show all your work to receive full credit for this problem.
Use the Substitution method to solve the system of equations. Give your answer as an ordered pair (x,y).


x + y = -5
x - y = 3

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%2B1%2Ay=-5
1%2Ax-1%2Ay=3

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=-5-1%2AxSubtract 1%2Ax from both sides

y=%28-5-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=-5-1%2Ax Now we've fully isolated y

Since y equals -5-1%2Ax we can substitute the expression -5-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B-1%2Ahighlight%28%28-5-1%2Ax%29%29=3 Replace y with -5-1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax-1%2A%28-5%29-1%28-1%29x=3 Distribute -1 to -5-1%2Ax

1%2Ax%2B5%2B1%2Ax=3 Multiply



1%2Ax%2B5%2B1%2Ax=3 Reduce any fractions

1%2Ax%2B1%2Ax=3-5 Subtract 5 from both sides


1%2Ax%2B1%2Ax=-2 Combine the terms on the right side



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


cross%28%281%2F2%29%282%2F1%29%29x=%28-2%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply -2%2F1 and 1%2F2 (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

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

-1-1%2Ay=3 Multiply

-1%2Ay=3%2B1Add 1 to both sides

-1%2Ay=4 Combine the terms on the right side

cross%28%281%2F-1%29%28-1%29%29%2Ay=%284%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1 on the left side.

y=4%2F-1 Multiply the terms on the right side


y=-4 Reduce


So this is the other answer


y=-4<---------------------------------Other answer


So our solution is

x=-1 and y=-4

which can also look like

(-1,-4)

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

1%2Ax%2B1%2Ay=-5
1%2Ax-1%2Ay=3

we get


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


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Check:

Plug in (-1,-4) into the system of equations


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

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


-1-4=-5 Multiply


-5=-5 Add


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


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



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

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


-1%2B4=3 Multiply


3=3 Add


3=3 Reduce. Since this equation is true the solution works.


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


Since the solution (-1,-4) satisfies the system of equations


1%2Ax%2B1%2Ay=-5
1%2Ax-1%2Ay=3


this verifies our answer.