SOLUTION: -x + y =2 x + y =4

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Question 117992: -x + y =2
x + y =4
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=2
1%2Ax%2B1%2Ay=4

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=2%2B1%2AxAdd 1%2Ax to both sides

y=%282%2B1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=2%2B1%2Ax Now we've fully isolated y

Since y equals 2%2B1%2Ax we can substitute the expression 2%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%282%2B1%2Ax%29%29=4 Replace y with 2%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%282%29%2B1%281%29x=4 Distribute 1 to 2%2B1%2Ax

1%2Ax%2B2%2B1%2Ax=4 Multiply



1%2Ax%2B2%2B1%2Ax=4 Reduce any fractions

1%2Ax%2B1%2Ax=4-2 Subtract 2 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=%282%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%281%29%2B1%2Ay=4 Plug in x=1 into the 2nd equation

1%2B1%2Ay=4 Multiply

1%2Ay=4-1Subtract 1 from both sides

1%2Ay=3 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%283%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=3%2F1 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=1 and y=3

which can also look like

(1,3)

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

-1%2Ax%2B1%2Ay=2
1%2Ax%2B1%2Ay=4

we get


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


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Check:

Plug in (1,3) into the system of equations


Let x=1 and y=3. Now plug those values into the equation -1%2Ax%2B1%2Ay=2

-1%2A%281%29%2B1%2A%283%29=2 Plug in x=1 and y=3


-1%2B3=2 Multiply


2=2 Add


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


So the solution (1,3) satisfies -1%2Ax%2B1%2Ay=2



Let x=1 and y=3. Now plug those values into the equation 1%2Ax%2B1%2Ay=4

1%2A%281%29%2B1%2A%283%29=4 Plug in x=1 and y=3


1%2B3=4 Multiply


4=4 Add


4=4 Reduce. Since this equation is true the solution works.


So the solution (1,3) satisfies 1%2Ax%2B1%2Ay=4


Since the solution (1,3) satisfies the system of equations


-1%2Ax%2B1%2Ay=2
1%2Ax%2B1%2Ay=4


this verifies our answer.