SOLUTION: please help solve using elimination method 3r-5s=-15 5r+3s=43

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Question 405203: please help
solve using elimination method
3r-5s=-15
5r+3s=43
Found 2 solutions by stanbon, MathLover1:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
solve using elimination method
3r-5s=-15
5r+3s=43
--------
Multiply 1st by 5 and 2nd by 3 to get:
15r - 25s = -75
15r + 9s = 129
---------
Subtract the top from the bottom and solve for "s":
34s = 204
s = 6
----
Solve for "r":
3r-5s = -15
3r - 5*6 = -15
3r = 15
r = 5
=======
Solution: r = 5; s=6
==========================
Cheers,
Stan H.
===========

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

3r-5s=-15
5r%2B3s=43


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=-15
5%2Ax%2B3%2Ay=43

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=%28-15%29%2A5 Multiply the top equation (both sides) by 5
-3%2A%285%2Ax%2B3%2Ay%29=%2843%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
15%2Ax-25%2Ay=-75
-15%2Ax-9%2Ay=-129

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-9%2Ay%29=-75-129

%2815-15%29%2Ax-25-9%29y=-75-129

cross%2815%2B-15%29%2Ax%2B%28-25-9%29%2Ay=-75-129 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:

-34%2Ay=-204

y=-204%2F-34 Divide both sides by -34 to solve for y



y=6 Reduce


Now plug this answer into the top equation 3%2Ax-5%2Ay=-15 to solve for x

3%2Ax-5%286%29=-15 Plug in y=6


3%2Ax-30=-15 Multiply



3%2Ax=-15%2B30 Subtract -30 from both sides

3%2Ax=15 Combine the terms on the right side

cross%28%281%2F3%29%283%29%29%2Ax=%2815%29%281%2F3%29 Multiply both sides by 1%2F3. This will cancel out 3 on the left side.


x=5 Multiply the terms on the right side


So our answer is

x=5, y=6

which also looks like

(5, 6)

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

3%2Ax-5%2Ay=-15
5%2Ax%2B3%2Ay=43

we get



graph of 3%2Ax-5%2Ay=-15 (red) 5%2Ax%2B3%2Ay=43 (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 (5,6). This verifies our answer.



your r=x and s=y
so,
r=5 and s=6