SOLUTION: Can someone help me solve by the elimination method. 5r – 4s =2 4r +5s = 18 What is the solution?

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Question 462344: Can someone help me solve by the elimination method.
5r – 4s =2
4r +5s = 18
What is the solution?
Answer by MathLover1(20850) 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

5%2Ax-4%2Ay=2
4%2Ax%2B5%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 5 and 4 to some equal number, we could try to get them to the LCM.

Since the LCM of 5 and 4 is 20, we need to multiply both sides of the top equation by 4 and multiply both sides of the bottom equation by -5 like this:

4%2A%285%2Ax-4%2Ay%29=%282%29%2A4 Multiply the top equation (both sides) by 4
-5%2A%284%2Ax%2B5%2Ay%29=%2818%29%2A-5 Multiply the bottom equation (both sides) by -5


So after multiplying we get this:
20%2Ax-16%2Ay=8
-20%2Ax-25%2Ay=-90

Notice how 20 and -20 add to zero (ie 20%2B-20=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2820%2Ax-20%2Ax%29-16%2Ay-25%2Ay%29=8-90

%2820-20%29%2Ax-16-25%29y=8-90

cross%2820%2B-20%29%2Ax%2B%28-16-25%29%2Ay=8-90 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:

-41%2Ay=-82

y=-82%2F-41 Divide both sides by -41 to solve for y



y=2 Reduce


Now plug this answer into the top equation 5%2Ax-4%2Ay=2 to solve for x

5%2Ax-4%282%29=2 Plug in y=2


5%2Ax-8=2 Multiply



5%2Ax=2%2B8 Subtract -8 from both sides

5%2Ax=10 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ax=%2810%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=2

which also looks like

(2, 2)

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

5%2Ax-4%2Ay=2
4%2Ax%2B5%2Ay=18

we get



graph of 5%2Ax-4%2Ay=2 (red) 4%2Ax%2B5%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 (2,2). This verifies our answer.