SOLUTION: Solve by addition: 4x-7y = 19 x+4y= -1

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Question 110665: Solve by addition:
4x-7y = 19
x+4y= -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

4%2Ax-7%2Ay=19
1%2Ax%2B4%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 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%284%2Ax-7%2Ay%29=%2819%29%2A1 Multiply the top equation (both sides) by 1
-4%2A%281%2Ax%2B4%2Ay%29=%28-1%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
4%2Ax-7%2Ay=19
-4%2Ax-16%2Ay=4

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


Now add the equations together. In order to add 2 equations, group like terms and combine them
%284%2Ax-4%2Ax%29-7%2Ay-16%2Ay%29=19%2B4

%284-4%29%2Ax-7-16%29y=19%2B4

cross%284%2B-4%29%2Ax%2B%28-7-16%29%2Ay=19%2B4 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:

-23%2Ay=23

y=23%2F-23 Divide both sides by -23 to solve for y



y=-1 Reduce


Now plug this answer into the top equation 4%2Ax-7%2Ay=19 to solve for x

4%2Ax-7%28-1%29=19 Plug in y=-1


4%2Ax%2B7=19 Multiply



4%2Ax=19-7 Subtract 7 from both sides

4%2Ax=12 Combine the terms on the right side

cross%28%281%2F4%29%284%29%29%2Ax=%2812%29%281%2F4%29 Multiply both sides by 1%2F4. This will cancel out 4 on the left side.


x=3 Multiply the terms on the right side


So our answer is

x=3, y=-1

which also looks like

(3, -1)

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

4%2Ax-7%2Ay=19
1%2Ax%2B4%2Ay=-1

we get



graph of 4%2Ax-7%2Ay=19 (red) 1%2Ax%2B4%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 (3,-1). This verifies our answer.