SOLUTION: Please help. Find values for m and b in the following system so that the solution to the system is (-3, 4). 5x = 7y =b mx + y = 22 Thanks JMS

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Question 113587: Please help.
Find values for m and b in the following system so that the solution to the system is (-3, 4).
5x = 7y =b
mx + y = 22
Thanks
JMS
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Find values for m and b in the following system so that the solution to the system is (-3, 4).
Given:
5x+%2B+7y+=b
mx+%2B+y+=+22
x+=-3
y+=+4
So:
5x+%2B+7y+=b
5%28-3%29+%2B+7%2A4+=b
-15+%2B+28+=b
13+=b
Or
b+=+13

mx+%2B+y+=+22
-3m+%2B+4+=+22

-3m+=+22+-+4

-3m+=++18

m+=++18%2F-3

m+=++-+6
then we have system:
5x+%2B+7y+=+13
-6x+%2B+y+=+22
and its solution is:
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

5%2Ax%2B7%2Ay=13
-6%2Ax%2B1%2Ay=22

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 5 and -6 to some equal number, we could try to get them to the LCM.

Since the LCM of 5 and -6 is -30, we need to multiply both sides of the top equation by -6 and multiply both sides of the bottom equation by -5 like this:

-6%2A%285%2Ax%2B7%2Ay%29=%2813%29%2A-6 Multiply the top equation (both sides) by -6
-5%2A%28-6%2Ax%2B1%2Ay%29=%2822%29%2A-5 Multiply the bottom equation (both sides) by -5


So after multiplying we get this:
-30%2Ax-42%2Ay=-78
30%2Ax-5%2Ay=-110

Notice how -30 and 30 add to zero (ie -30%2B30=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%28-30%2Ax%2B30%2Ax%29-42%2Ay-5%2Ay%29=-78-110

%28-30%2B30%29%2Ax-42-5%29y=-78-110

cross%28-30%2B30%29%2Ax%2B%28-42-5%29%2Ay=-78-110 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:

-47%2Ay=-188

y=-188%2F-47 Divide both sides by -47 to solve for y



y=4 Reduce


Now plug this answer into the top equation 5%2Ax%2B7%2Ay=13 to solve for x

5%2Ax%2B7%284%29=13 Plug in y=4


5%2Ax%2B28=13 Multiply



5%2Ax=13-28 Subtract 28 from both sides

5%2Ax=-15 Combine the terms on the right side

cross%28%281%2F5%29%285%29%29%2Ax=%28-15%29%281%2F5%29 Multiply both sides by 1%2F5. This will cancel out 5 on the left side.


x=-3 Multiply the terms on the right side


So our answer is

x=-3, y=4

which also looks like

(-3, 4)

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

5%2Ax%2B7%2Ay=13
-6%2Ax%2B1%2Ay=22

we get



graph of 5%2Ax%2B7%2Ay=13 (red) -6%2Ax%2B1%2Ay=22 (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 (-3,4). This verifies our answer.