SOLUTION: Use elimination to solve: x/2+2y=9 2x-y=18

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Question 112087: Use elimination to solve: x/2+2y=9
2x-y=18
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
TEST

%281%2F2%29%2Ax%2B%281%29%2Ay=9 Start with the first equation


2%28%281%2F2%29%2Ax%2B%281%29%2Ay%29=%282%29%2A%289%29 Multiply both sides by the LCD 2



1%2Ax%2B2%2Ay=18Distribute and simplify


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Lets start with the given system of linear equations

1%2Ax%2B2%2Ay=18
2%2Ax-1%2Ay=18

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

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

2%2A%281%2Ax%2B2%2Ay%29=%2818%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%282%2Ax-1%2Ay%29=%2818%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
2%2Ax%2B4%2Ay=36
-2%2Ax%2B1%2Ay=-18

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


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%284%2Ay%2B1%2Ay%29=36-18

%282-2%29%2Ax%2B%284%2B1%29y=36-18

cross%282%2B-2%29%2Ax%2B%284%2B1%29%2Ay=36-18 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:

5%2Ay=18

y=18%2F5 Divide both sides by 5 to solve for y



y=18%2F5 Reduce


Now plug this answer into the top equation 1%2Ax%2B2%2Ay=18 to solve for x

1%2Ax%2B2%2818%2F5%29=18 Plug in y=18%2F5


1%2Ax%2B36%2F5=18 Multiply



1%2Ax%2B36%2F5=18 Reduce



1%2Ax=18-36%2F5 Subtract 36%2F5 from both sides

1%2Ax=90%2F5-36%2F5 Make 18 into a fraction with a denominator of 5

1%2Ax=54%2F5 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2854%2F5%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=54%2F5 Multiply the terms on the right side


So our answer is

x=54%2F5, y=18%2F5

which also looks like

(54%2F5, 18%2F5)

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

1%2Ax%2B2%2Ay=18
2%2Ax-1%2Ay=18

we get



graph of 1%2Ax%2B2%2Ay=18 (red) 2%2Ax-1%2Ay=18 (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 (54%2F5,18%2F5). This verifies our answer.