SOLUTION: Solve the system by addition. 5x – 3y = 13 4x – 3y = 11

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Question 117213: Solve the system by addition.

5x – 3y = 13
4x – 3y = 11
Found 2 solutions by jim_thompson5910, checkley71:
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

5%2Ax-3%2Ay=13
4%2Ax-3%2Ay=11

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-3%2Ay%29=%2813%29%2A4 Multiply the top equation (both sides) by 4
-5%2A%284%2Ax-3%2Ay%29=%2811%29%2A-5 Multiply the bottom equation (both sides) by -5


So after multiplying we get this:
20%2Ax-12%2Ay=52
-20%2Ax%2B15%2Ay=-55

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%29-12%2Ay%2B15%2Ay%29=52-55

%2820-20%29%2Ax-12%2B15%29y=52-55

cross%2820%2B-20%29%2Ax%2B%28-12%2B15%29%2Ay=52-55 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:

3%2Ay=-3

y=-3%2F3 Divide both sides by 3 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 5%2Ax-3%2Ay=13 to solve for x

5%2Ax-3%28-1%29=13 Plug in y=-1


5%2Ax%2B3=13 Multiply



5%2Ax=13-3 Subtract 3 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=-1

which also looks like

(2, -1)

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

5%2Ax-3%2Ay=13
4%2Ax-3%2Ay=11

we get



graph of 5%2Ax-3%2Ay=13 (red) 4%2Ax-3%2Ay=11 (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,-1). This verifies our answer.

Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
5x-3y=13 now multiply the other equation by -1.
-4x+3y=-11 now add.
-----------------------
x=2 answer.
5*2-3y=13
10-3y=13
-3y=13-10
-3y=3
y=y=3/-3
y=-1 answer.