SOLUTION: solve the following systems of equations by elimination. express the solution an an ordered pair 3x+3y=-15 5x+4y=-18

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Question 1171070: solve the following systems of equations by elimination. express the solution an an ordered pair
3x+3y=-15
5x+4y=-18
Found 2 solutions by MathLover1, Solver92311:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

3x%2B3y=-15
5x%2B4y=-18
--------------------

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


Lets start with the given system of linear equations

3%2Ax%2B3%2Ay=-15
5%2Ax%2B4%2Ay=-18

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%2B3%2Ay%29=%28-15%29%2A5 Multiply the top equation (both sides) by 5
-3%2A%285%2Ax%2B4%2Ay%29=%28-18%29%2A-3 Multiply the bottom equation (both sides) by -3


So after multiplying we get this:
15%2Ax%2B15%2Ay=-75
-15%2Ax-12%2Ay=54

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%2B%2815%2Ay-12%2Ay%29=-75%2B54

%2815-15%29%2Ax%2B%2815-12%29y=-75%2B54

cross%2815%2B-15%29%2Ax%2B%2815-12%29%2Ay=-75%2B54 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:

3%2Ay=-21

y=-21%2F3 Divide both sides by 3 to solve for y



y=-7 Reduce


Now plug this answer into the top equation 3%2Ax%2B3%2Ay=-15 to solve for x

3%2Ax%2B3%28-7%29=-15 Plug in y=-7


3%2Ax-21=-15 Multiply



3%2Ax=-15%2B21 Subtract -21 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=-7

which also looks like

(2, -7)

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

3%2Ax%2B3%2Ay=-15
5%2Ax%2B4%2Ay=-18

we get



graph of 3%2Ax%2B3%2Ay=-15 (red) 5%2Ax%2B4%2Ay=-18 (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,-7). This verifies our answer.

Answer by Solver92311(821) About Me  (Show Source):
You can put this solution on YOUR website!


Multiply the first equation by and the second equation by . Then add the equations term-by-term. The -terms will be eliminated leaving you with one equation in . Solve for and then substitute that value back into either one of the original equations and solve for . I sincerely hope you don't need instructions on how to create an ordered pair.


John

My calculator said it, I believe it, that settles it

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