SOLUTION: Solve the system by addition. 4x – 9y = –14 x + 2y = 5

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Question 114179: Solve the system by addition.
4x – 9y = –14
x + 2y = 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-9%2Ay=-14
1%2Ax%2B2%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%284%2Ax-9%2Ay%29=%28-14%29%2A1 Multiply the top equation (both sides) by 1
-4%2A%281%2Ax%2B2%2Ay%29=%285%29%2A-4 Multiply the bottom equation (both sides) by -4


So after multiplying we get this:
4%2Ax-9%2Ay=-14
-4%2Ax-8%2Ay=-20

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%29-9%2Ay-8%2Ay%29=-14-20

%284-4%29%2Ax-9-8%29y=-14-20

cross%284%2B-4%29%2Ax%2B%28-9-8%29%2Ay=-14-20 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:

-17%2Ay=-34

y=-34%2F-17 Divide both sides by -17 to solve for y



y=2 Reduce


Now plug this answer into the top equation 4%2Ax-9%2Ay=-14 to solve for x

4%2Ax-9%282%29=-14 Plug in y=2


4%2Ax-18=-14 Multiply



4%2Ax=-14%2B18 Subtract -18 from both sides

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

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


x=1 Multiply the terms on the right side


So our answer is

x=1, y=2

which also looks like

(1, 2)

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

4%2Ax-9%2Ay=-14
1%2Ax%2B2%2Ay=5

we get



graph of 4%2Ax-9%2Ay=-14 (red) 1%2Ax%2B2%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 (1,2). This verifies our answer.