SOLUTION: Solve each system by addition x+y=7 x-y=9

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Question 177969: Solve each system by addition
x+y=7
x-y=9
Found 2 solutions by jim_thompson5910, Mathtut:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

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

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 1 is 1, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -1 like this:

1%2A%281%2Ax%2B1%2Ay%29=%287%29%2A1 Multiply the top equation (both sides) by 1
-1%2A%281%2Ax-1%2Ay%29=%289%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
1%2Ax%2B1%2Ay=7
-1%2Ax%2B1%2Ay=-9

Notice how 1 and -1 add to zero (ie 1%2B-1=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%281%2Ax-1%2Ax%29%2B%281%2Ay%2B1%2Ay%29=7-9

%281-1%29%2Ax%2B%281%2B1%29y=7-9

cross%281%2B-1%29%2Ax%2B%281%2B1%29%2Ay=7-9 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

2%2Ay=-2

y=-2%2F2 Divide both sides by 2 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=7 to solve for x

1%2Ax%2B1%28-1%29=7 Plug in y=-1


1%2Ax-1=7 Multiply



1%2Ax=7%2B1 Subtract -1 from both sides

1%2Ax=8 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%288%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=8 Multiply the terms on the right side


So our answer is

x=8, y=-1

which also looks like

(8, -1)

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

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

we get



graph of 1%2Ax%2B1%2Ay=7 (red) 1%2Ax-1%2Ay=9 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (8,-1). This verifies our answer.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
addition is really just another name for elimination
:
these two equations are already set up and do not have to be manipulated at all because one of the variables will be eliminated just by adding the equations together.
:
x+y=7....eq 1
x-y=9....eq 2
:
notice as these are lined up that when we add the two equations together the y terms are eliminated? why? because +y-y=0. We are left with x+x=7+9
:
2x=16
:
highlight%28x=8%29
:
now take x's found value of 8 and plug it into either eq 1 or 2 and solve for y. I chose eq 1
:
8+y=7
:
highlight%28y=-1%29
:
so the solution(8,-1) whould work in both equations
:
x+y=7....eq 1--->8+(-1)=7--->7=7....check
x-y=9....eq 2--->8-(-1)=9--->9=9....check