SOLUTION: 21. Given: Ray ST bisects Angle(<) RSV m< RST= x+y m< TSV= 2x-2y m< RSV= 64 degrees Find: x and y It also gives a diagram before it which

Algebra ->  Geometry-proofs -> SOLUTION: 21. Given: Ray ST bisects Angle(<) RSV m< RST= x+y m< TSV= 2x-2y m< RSV= 64 degrees Find: x and y It also gives a diagram before it which      Log On


   

Question 85400: 21. Given: Ray ST bisects Angle(<) RSV
m< RST= x+y
m< TSV= 2x-2y
m< RSV= 64 degrees
Find: x and y
It also gives a diagram before it which i will try and describe:
It is a pic of an angle split in two. the vertex is named s and the top ray has the endpoint called r and the middle endpoint it t and the bottom endpoint is v. I have tried to explain the diagram but i don't think i did it very well. oh, well.
if you go to this picture site and it has the picture but just different names for the points: http://education.yahoo.com/homework_help/math_help/solutionimages/minigeogt/1/1/1/minigeogt_1_1_1_13_1/f-37-49-oyo-1.gif
I'm sorry if you are asking what in the world is she talking about because it is understandable!
thank you if you find the answer!
-Athena
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Since m< RST and m< TSV are half of m< RSV, we know this expression is true:

m%3CRST=%281%2F2%29%2A64=32
m%3CTSV=%281%2F2%29%2A64=32
Also, since m< RST= x+y and m< TSV= 2x-2y, we can set up the following system of equations:

x%2By=32
2x-2y=32

Now we can solve for x and y by addition/elimination:

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


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=32
2%2Ax-2%2Ay=32

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

Since the LCM of 1 and 2 is 2, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:

2%2A%281%2Ax%2B1%2Ay%29=%2832%29%2A2 Multiply the top equation (both sides) by 2
-1%2A%282%2Ax-2%2Ay%29=%2832%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
2%2Ax%2B2%2Ay=64
-2%2Ax%2B2%2Ay=-32

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


Now add the equations together. In order to add 2 equations, group like terms and combine them
%282%2Ax-2%2Ax%29%2B%282%2Ay%2B2%2Ay%29=64-32

%282-2%29%2Ax%2B%282%2B2%29y=64-32

cross%282%2B-2%29%2Ax%2B%282%2B2%29%2Ay=64-32 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:

4%2Ay=32

y=32%2F4 Divide both sides by 4 to solve for y



y=8 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=32 to solve for x

1%2Ax%2B1%288%29=32 Plug in y=8


1%2Ax%2B8=32 Multiply



1%2Ax=32-8 Subtract 8 from both sides

1%2Ax=24 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2824%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=24 Multiply the terms on the right side


So our answer is

x=24, y=8

which also looks like

(24, 8)

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

1%2Ax%2B1%2Ay=32
2%2Ax-2%2Ay=32

we get



graph of 1%2Ax%2B1%2Ay=32 (red) 2%2Ax-2%2Ay=32 (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 (24,8). This verifies our answer.