SOLUTION: Solve the system by addition. 2x - 3y = 2 x - 5y = 1

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Question 84927: Solve the system by addition.
2x - 3y = 2
x - 5y = 1
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

2%2Ax-3%2Ay=2
1%2Ax-5%2Ay=1

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

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

1%2A%282%2Ax-3%2Ay%29=%282%29%2A1 Multiply the top equation (both sides) by 1
-2%2A%281%2Ax-5%2Ay%29=%281%29%2A-2 Multiply the bottom equation (both sides) by -2


So after multiplying we get this:
2%2Ax-3%2Ay=2
-2%2Ax%2B10%2Ay=-2

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%29-3%2Ay%2B10%2Ay%29=2-2

%282-2%29%2Ax-3%2B10%29y=2-2

cross%282%2B-2%29%2Ax%2B%28-3%2B10%29%2Ay=2-2 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:

7%2Ay=0

y=0%2F7 Divide both sides by 7 to solve for y



y=0 Reduce


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

2%2Ax-3%280%29=2 Plug in y=0


2%2Ax%2B0=2 Multiply



2%2Ax=2-0 Subtract 0 from both sides

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

cross%28%281%2F2%29%282%29%29%2Ax=%282%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2 on the left side.


x=1 Multiply the terms on the right side


So our answer is

x=1, y=0

which also looks like

(1, 0)

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

2%2Ax-3%2Ay=2
1%2Ax-5%2Ay=1

we get



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