Questions on Geometry: Angles, complementary, supplementary angles answered by real tutors!

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Question 147652: The sum of the measures of two obtuse angle is 215.The sum of three-fifths of the supplement of the smaller angle and two-thirds the supplement of the larger angle is 91. Find the measure of the two angles.: The sum of the measures of two obtuse angle is 215.The sum of three-fifths of the supplement of the smaller angle and two-thirds the supplement of the larger angle is 91. Find the measure of the two angles.
Answer by vleith(1238) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two angels be a and b
a+b = 215
We are also given a second set of information
The supplement of an angle plus the angle is always 180.
We are told (3/5)*(180-a) + (2/3)*(180-b) = 91
108 - (3/5)a + 120 - (2/3)b = 91
-(3/5)a - (2/3)b = -137
9a + 10b = 2055
Now solve using
9a + 10b = 2055
a + b = 215
Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

9*x+10*y=2055
1*x+1*y=215

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

Since the LCM of 9 and 1 is 9, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -9 like this:

1*(9*x+10*y)=(2055)*1 Multiply the top equation (both sides) by 1
-9*(1*x+1*y)=(215)*-9 Multiply the bottom equation (both sides) by -9


So after multiplying we get this:
9*x+10*y=2055
-9*x-9*y=-1935

Notice how 9 and -9 add to zero (ie 9+-9=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
(9*x-9*x)+(10*y-9*y)=2055-1935

(9-9)*x+(10-9)y=2055-1935

cross(9+-9)*x+(10-9)*y=2055-1935 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:

1*y=120

y=120 Divide both sides by 1 to solve for y



y=120 Reduce


Now plug this answer into the top equation 9*x+10*y=2055 to solve for x

9*x+10(120)=2055 Plug in y=120


9*x+1200=2055 Multiply



9*x=2055-1200 Subtract 1200 from both sides

9*x=855 Combine the terms on the right side

cross((1/9)(9))*x=(855)(1/9) Multiply both sides by 1/9. This will cancel out 9 on the left side.


x=95 Multiply the terms on the right side


So our answer is

x=95, y=120

which also looks like

(95, 120)

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

9*x+10*y=2055
1*x+1*y=215

we get



<BR> drawing( 500, 600, -105, 105, -130, 130,<BR> graph( 500, 600, -105, 105, -130, 130, (2055-9*x)/10, (215-1*x)/1 ),<BR> blue( circle( 95, 120, 1.4 ) ) <BR> )<BR> graph of 9*x+10*y=2055 (red) 1*x+1*y=215 (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 (95,120). This verifies our answer.


To get a = 95 and b = 120