SOLUTION: HOW DO YOU DO THIS ONE USING ELIMINATION OR SUBSTITUTION....9X-3Y=9 AND X+3Y=11

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Question 120421: HOW DO YOU DO THIS ONE USING ELIMINATION OR SUBSTITUTION....9X-3Y=9 AND X+3Y=11
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

9%2Ax-3%2Ay=9
1%2Ax%2B3%2Ay=11

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 9 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of 9 and 1 is 9, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -9 like this:

1%2A%289%2Ax-3%2Ay%29=%289%29%2A1 Multiply the top equation (both sides) by 1
-9%2A%281%2Ax%2B3%2Ay%29=%2811%29%2A-9 Multiply the bottom equation (both sides) by -9


So after multiplying we get this:
9%2Ax-3%2Ay=9
-9%2Ax-27%2Ay=-99

Notice how 9 and -9 add to zero (ie 9%2B-9=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%289%2Ax-9%2Ax%29-3%2Ay-27%2Ay%29=9-99

%289-9%29%2Ax-3-27%29y=9-99

cross%289%2B-9%29%2Ax%2B%28-3-27%29%2Ay=9-99 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:

-30%2Ay=-90

y=-90%2F-30 Divide both sides by -30 to solve for y



y=3 Reduce


Now plug this answer into the top equation 9%2Ax-3%2Ay=9 to solve for x

9%2Ax-3%283%29=9 Plug in y=3


9%2Ax-9=9 Multiply



9%2Ax=9%2B9 Subtract -9 from both sides

9%2Ax=18 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ax=%2818%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=3

which also looks like

(2, 3)

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

9%2Ax-3%2Ay=9
1%2Ax%2B3%2Ay=11

we get



graph of 9%2Ax-3%2Ay=9 (red) 1%2Ax%2B3%2Ay=11 (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 (2,3). This verifies our answer.