SOLUTION: solve the system by substitution x + y = 12 -x + y = 2

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Question 92333: solve the system by substitution
x + y = 12
-x + y = 2
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=12
-1%2Ax%2B1%2Ay=2

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

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


Which breaks down and reduces to



y=12-1%2Ax Now we've fully isolated y

Since y equals 12-1%2Ax we can substitute the expression 12-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


-1%2Ax%2B1%2Ahighlight%28%2812-1%2Ax%29%29=2 Replace y with 12-1%2Ax. Since this eliminates y, we can now solve for x.

-1%2Ax%2B1%2A%2812%29%2B1%28-1%29x=2 Distribute 1 to 12-1%2Ax

-1%2Ax%2B12-1%2Ax=2 Multiply



-1%2Ax%2B12-1%2Ax=2 Reduce any fractions

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


-1%2Ax-1%2Ax=-10 Combine the terms on the right side



-2%2Ax=-10 Now combine the terms on the left side.


cross%28%281%2F-2%29%28-2%2F1%29%29x=%28-10%2F1%29%281%2F-2%29 Multiply both sides by 1%2F-2. This will cancel out -2%2F1 and isolate x

So when we multiply -10%2F1 and 1%2F-2 (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

-1%285%29%2B1%2Ay=2 Plug in x=5 into the 2nd equation

-5%2B1%2Ay=2 Multiply

1%2Ay=2%2B5Add 5 to both sides

1%2Ay=7 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%287%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=7%2F1 Multiply the terms on the right side


y=7 Reduce


So this is the other answer


y=7<---------------------------------Other answer


So our solution is

x=5 and y=7

which can also look like

(5,7)

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

1%2Ax%2B1%2Ay=12
-1%2Ax%2B1%2Ay=2

we get


graph of 1%2Ax%2B1%2Ay=12 (red) and -1%2Ax%2B1%2Ay=2 (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 (5,7). This verifies our answer.


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Check:

Plug in (5,7) into the system of equations


Let x=5 and y=7. Now plug those values into the equation 1%2Ax%2B1%2Ay=12

1%2A%285%29%2B1%2A%287%29=12 Plug in x=5 and y=7


5%2B7=12 Multiply


12=12 Add


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


So the solution (5,7) satisfies 1%2Ax%2B1%2Ay=12



Let x=5 and y=7. Now plug those values into the equation -1%2Ax%2B1%2Ay=2

-1%2A%285%29%2B1%2A%287%29=2 Plug in x=5 and y=7


-5%2B7=2 Multiply


2=2 Add


2=2 Reduce. Since this equation is true the solution works.


So the solution (5,7) satisfies -1%2Ax%2B1%2Ay=2


Since the solution (5,7) satisfies the system of equations


1%2Ax%2B1%2Ay=12
-1%2Ax%2B1%2Ay=2


this verifies our answer.