SOLUTION: How can I solve 3a + 5b = 31 7a + 5b = 59 Using elimination? (or linear combination method) Thanks!

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Question 120417: How can I solve 3a + 5b = 31
7a + 5b = 59

Using elimination? (or linear combination method)

Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
note: I'm going to use x and y instead of a and b

Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

3%2Ax%2B5%2Ay=31
7%2Ax%2B5%2Ay=59

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 7 to some equal number, we could try to get them to the LCM.

Since the LCM of 3 and 7 is 21, we need to multiply both sides of the top equation by 7 and multiply both sides of the bottom equation by -3 like this:

7%2A%283%2Ax%2B5%2Ay%29=%2831%29%2A7 Multiply the top equation (both sides) by 7
-3%2A%287%2Ax%2B5%2Ay%29=%2859%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
21%2Ax%2B35%2Ay=217
-21%2Ax-15%2Ay=-177

Notice how 21 and -21 add to zero (ie 21%2B-21=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2821%2Ax-21%2Ax%29%2B%2835%2Ay-15%2Ay%29=217-177

%2821-21%29%2Ax%2B%2835-15%29y=217-177

cross%2821%2B-21%29%2Ax%2B%2835-15%29%2Ay=217-177 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:

20%2Ay=40

y=40%2F20 Divide both sides by 20 to solve for y



y=2 Reduce


Now plug this answer into the top equation 3%2Ax%2B5%2Ay=31 to solve for x

3%2Ax%2B5%282%29=31 Plug in y=2


3%2Ax%2B10=31 Multiply



3%2Ax=31-10 Subtract 10 from both sides

3%2Ax=21 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2821%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=7 Multiply the terms on the right side


So our answer is

x=7, y=2

which also looks like

(7, 2)

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

3%2Ax%2B5%2Ay=31
7%2Ax%2B5%2Ay=59

we get



graph of 3%2Ax%2B5%2Ay=31 (red) 7%2Ax%2B5%2Ay=59 (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 (7,2). This verifies our answer.