SOLUTION: Solve using the elimination method: 3x-5y=-4 5x-3y=4

Algebra ->  Matrices-and-determiminant -> SOLUTION: Solve using the elimination method: 3x-5y=-4 5x-3y=4      Log On


   

Question 104138: Solve using the elimination method:
3x-5y=-4
5x-3y=4
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

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

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 3 and 5 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 5 is 15, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -3 like this:

5%2A%283%2Ax-5%2Ay%29=%28-4%29%2A5 Multiply the top equation (both sides) by 5
-3%2A%285%2Ax-3%2Ay%29=%284%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
15%2Ax-25%2Ay=-20
-15%2Ax%2B9%2Ay=-12

Notice how 15 and -15 add to zero (ie 15%2B-15=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2815%2Ax-15%2Ax%29-25%2Ay%2B9%2Ay%29=-20-12

%2815-15%29%2Ax-25%2B9%29y=-20-12

cross%2815%2B-15%29%2Ax%2B%28-25%2B9%29%2Ay=-20-12 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:

-16%2Ay=-32

y=-32%2F-16 Divide both sides by -16 to solve for y



y=2 Reduce


Now plug this answer into the top equation 3%2Ax-5%2Ay=-4 to solve for x

3%2Ax-5%282%29=-4 Plug in y=2


3%2Ax-10=-4 Multiply



3%2Ax=-4%2B10 Subtract -10 from both sides

3%2Ax=6 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%286%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=2 Multiply the terms on the right side


So our answer is

x=2, y=2

which also looks like

(2, 2)

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

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

we get



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