SOLUTION: Solve the system of equations using the addition (elimination) method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answe

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Question 107969: Solve the system of equations using the addition (elimination) method.
If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is “no solution” or “infinitely many solutions.”
9x + 2y = 2
3x + 5y = 5
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%2B2%2Ay=2
3%2Ax%2B5%2Ay=5

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 3 to some equal number, we could try to get them to the LCM.

Since the LCM of 9 and 3 is 9, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:

1%2A%289%2Ax%2B2%2Ay%29=%282%29%2A1 Multiply the top equation (both sides) by 1
-3%2A%283%2Ax%2B5%2Ay%29=%285%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
9%2Ax%2B2%2Ay=2
-9%2Ax-15%2Ay=-15

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%2B%282%2Ay-15%2Ay%29=2-15

%289-9%29%2Ax%2B%282-15%29y=2-15

cross%289%2B-9%29%2Ax%2B%282-15%29%2Ay=2-15 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:

-13%2Ay=-13

y=-13%2F-13 Divide both sides by -13 to solve for y



y=1 Reduce


Now plug this answer into the top equation 9%2Ax%2B2%2Ay=2 to solve for x

9%2Ax%2B2%281%29=2 Plug in y=1


9%2Ax%2B2=2 Multiply



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

9%2Ax=0 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ax=%280%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.


x=0 Multiply the terms on the right side


So our answer is

x=0, y=1

which also looks like

(0, 1)

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

9%2Ax%2B2%2Ay=2
3%2Ax%2B5%2Ay=5

we get



graph of 9%2Ax%2B2%2Ay=2 (red) 3%2Ax%2B5%2Ay=5 (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 (0,1). This verifies our answer.