SOLUTION: Solve each system of equations. Check by substitution x - y = 3 x + y = -1

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Question 120286: Solve each system of equations. Check by substitution
x - y = 3
x + y = -1
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

x+-+y+=+3
x+%2B+y+=+-1
solution:
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

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

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=3-1%2AxSubtract 1%2Ax from both sides

y=%283-1%2Ax%29%2F-1 Divide both sides by -1.


Which breaks down and reduces to



y=-3%2B1%2Ax Now we've fully isolated y

Since y equals -3%2B1%2Ax we can substitute the expression -3%2B1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%28-3%2B1%2Ax%29%29=-1 Replace y with -3%2B1%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%28-3%29%2B1%281%29x=-1 Distribute 1 to -3%2B1%2Ax

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



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

1%2Ax%2B1%2Ax=-1%2B3Add 3 to 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=-1 Plug in x=1 into the 2nd equation

1%2B1%2Ay=-1 Multiply

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

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

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

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


y=-2 Reduce


So this is the other answer


y=-2<---------------------------------Other answer


So our solution is

x=1 and y=-2

which can also look like

(1,-2)

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

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

we get


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


-----------------------------------------------------------------------------------------------
Check:

Plug in (1,-2) into the system of equations


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

1%2A%281%29-1%2A%28-2%29=3 Plug in x=1 and y=-2


1%2B2=3 Multiply


3=3 Add


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


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



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

1%2A%281%29%2B1%2A%28-2%29=-1 Plug in x=1 and y=-2


1-2=-1 Multiply


-1=-1 Add


-1=-1 Reduce. Since this equation is true the solution works.


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


Since the solution (1,-2) satisfies the system of equations


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


this verifies our answer.