SOLUTION: I can't seem to come up with a correct answer to this one. Please help! Use the elimination method to solve each system. {{{m/9-n/13=1/117}}} {{{m/8-5n/16=19/16}}} I think I

Algebra ->  Linear-equations -> SOLUTION: I can't seem to come up with a correct answer to this one. Please help! Use the elimination method to solve each system. {{{m/9-n/13=1/117}}} {{{m/8-5n/16=19/16}}} I think I      Log On


   

Question 104558: I can't seem to come up with a correct answer to this one. Please help!
Use the elimination method to solve each system.
m%2F9-n%2F13=1%2F117
m%2F8-5n%2F16=19%2F16
I think I'm getting tripped up on the second equation.
I used the LCD to come up with
13m-9n=1
2m-5n=19
I ultimately end up with some obscure answer like 47m=166, which CAN'T be right.
Thanks!
Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
13m-9n=1
2m-5n=19
The above are correct.
Ed
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

13%2Ax-9%2Ay=1
2%2Ax-5%2Ay=19

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 13 and 2 to some equal number, we could try to get them to the LCM.

Since the LCM of 13 and 2 is 26, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -13 like this:

2%2A%2813%2Ax-9%2Ay%29=%281%29%2A2 Multiply the top equation (both sides) by 2
-13%2A%282%2Ax-5%2Ay%29=%2819%29%2A-13 Multiply the bottom equation (both sides) by -13


So after multiplying we get this:
26%2Ax-18%2Ay=2
-26%2Ax%2B65%2Ay=-247

Notice how 26 and -26 add to zero (ie 26%2B-26=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2826%2Ax-26%2Ax%29-18%2Ay%2B65%2Ay%29=2-247

%2826-26%29%2Ax-18%2B65%29y=2-247

cross%2826%2B-26%29%2Ax%2B%28-18%2B65%29%2Ay=2-247 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=-245

y=-245%2F47 Divide both sides by 47 to solve for y



y=-245%2F47 Reduce


Now plug this answer into the top equation 13%2Ax-9%2Ay=1 to solve for x

13%2Ax-9%28-245%2F47%29=1 Plug in y=-245%2F47


13%2Ax%2B2205%2F47=1 Multiply



13%2Ax%2B2205%2F47=1 Reduce



13%2Ax=1-2205%2F47 Subtract 2205%2F47 from both sides

13%2Ax=47%2F47-2205%2F47 Make 1 into a fraction with a denominator of 47

13%2Ax=-2158%2F47 Combine the terms on the right side

cross%28%281%2F13%29%2813%29%29%2Ax=%28-2158%2F47%29%281%2F13%29 Multiply both sides by 1%2F13. This will cancel out 13 on the left side.


x=-166%2F47 Multiply the terms on the right side


So our answer is

x=-166%2F47, y=-245%2F47

which also looks like

(-166%2F47, -245%2F47)

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

13%2Ax-9%2Ay=1
2%2Ax-5%2Ay=19

we get



graph of 13%2Ax-9%2Ay=1 (red) 2%2Ax-5%2Ay=19 (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 (-166%2F47,-245%2F47). This verifies our answer.