SOLUTION: Need to solve this by using elimination method. 2x+9y=3.75 x+y=6.5

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Question 945604: Need to solve this by using elimination method.
2x+9y=3.75
x+y=6.5
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

2x%2B9y=3.75 or 2x%2B9y=15%2F4 I will use this one, it will work better with this solver
x%2By=6.5 or x%2By=13%2F2

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


%282%29%2Ax%2B%289%29%2Ay=15%2F4 Start with the first equation


4%28%282%29%2Ax%2B%289%29%2Ay%29=%284%29%2A%2815%2F4%29 Multiply both sides by the LCD 4



8%2Ax%2B36%2Ay=15Distribute and simplify


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%281%29%2Ax%2B%281%29%2Ay=13%2F2 Start with the second equation


2%28%281%29%2Ax%2B%281%29%2Ay%29=%282%29%2A%2813%2F2%29 Multiply both sides by the LCD 2



2%2Ax%2B2%2Ay=13 Distribute and simplify



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Lets start with the given system of linear equations

8%2Ax%2B36%2Ay=15
2%2Ax%2B2%2Ay=13

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 8 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 8 and 2 is 8, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -4 like this:

1%2A%288%2Ax%2B36%2Ay%29=%2815%29%2A1 Multiply the top equation (both sides) by 1
-4%2A%282%2Ax%2B2%2Ay%29=%2813%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
8%2Ax%2B36%2Ay=15
-8%2Ax-8%2Ay=-52

Notice how 8 and -8 add to zero (ie 8%2B-8=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%288%2Ax-8%2Ax%29%2B%2836%2Ay-8%2Ay%29=15-52

%288-8%29%2Ax%2B%2836-8%29y=15-52

cross%288%2B-8%29%2Ax%2B%2836-8%29%2Ay=15-52 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:

28%2Ay=-37

y=-37%2F28 Divide both sides by 28 to solve for y



y=-37%2F28 Reduce


Now plug this answer into the top equation 8%2Ax%2B36%2Ay=15 to solve for x

8%2Ax%2B36%28-37%2F28%29=15 Plug in y=-37%2F28


8%2Ax-1332%2F28=15 Multiply



8%2Ax-333%2F7=15 Reduce



8%2Ax=15%2B333%2F7 Subtract -333%2F7 from both sides

8%2Ax=105%2F7%2B333%2F7 Make 15 into a fraction with a denominator of 7

8%2Ax=438%2F7 Combine the terms on the right side

cross%28%281%2F8%29%288%29%29%2Ax=%28438%2F7%29%281%2F8%29 Multiply both sides by 1%2F8. This will cancel out 8 on the left side.


x=219%2F28 Multiply the terms on the right side


So our answer is

x=219%2F28, y=-37%2F28

which also looks like

(219%2F28, -37%2F28)

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

8%2Ax%2B36%2Ay=15
2%2Ax%2B2%2Ay=13

we get



graph of 8%2Ax%2B36%2Ay=15 (red) 2%2Ax%2B2%2Ay=13 (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 (219%2F28,-37%2F28). This verifies our answer.



so, you can write your solution as decimal number like this: (7.8,-1.3)