SOLUTION: Solve the system by addition. –4x – 7y = –1, x – 3y = 5

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Question 116554: Solve the system by addition. –4x – 7y = –1, x – 3y = 5
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

-4%2Ax-7%2Ay=-1
1%2Ax-3%2Ay=5

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 -4 and 1 to some equal number, we could try to get them to the LCM.

Since the LCM of -4 and 1 is -4, 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%28-4%2Ax-7%2Ay%29=%28-1%29%2A1 Multiply the top equation (both sides) by 1
4%2A%281%2Ax-3%2Ay%29=%285%29%2A4 Multiply the bottom equation (both sides) by 4


So after multiplying we get this:
-4%2Ax-7%2Ay=-1
4%2Ax-12%2Ay=20

Notice how -4 and 4 add to zero (ie -4%2B4=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-4%2Ax%2B4%2Ax%29-7%2Ay-12%2Ay%29=-1%2B20

%28-4%2B4%29%2Ax-7-12%29y=-1%2B20

cross%28-4%2B4%29%2Ax%2B%28-7-12%29%2Ay=-1%2B20 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:

-19%2Ay=19

y=19%2F-19 Divide both sides by -19 to solve for y



y=-1 Reduce


Now plug this answer into the top equation -4%2Ax-7%2Ay=-1 to solve for x

-4%2Ax-7%28-1%29=-1 Plug in y=-1


-4%2Ax%2B7=-1 Multiply



-4%2Ax=-1-7 Subtract 7 from both sides

-4%2Ax=-8 Combine the terms on the right side

cross%28%281%2F-4%29%28-4%29%29%2Ax=%28-8%29%281%2F-4%29 Multiply both sides by 1%2F-4. This will cancel out -4 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)

-4%2Ax-7%2Ay=-1
1%2Ax-3%2Ay=5

we get



graph of -4%2Ax-7%2Ay=-1 (red) 1%2Ax-3%2Ay=5 (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.