Question

1. find the values of x and y in the diagram below. 6. find the values of x and y in the diagram below.

Explanation:

Step1: Solve for \(x\) in problem 5

Use the property of vertical angles (they are equal).
\(6x - 2+72=11x - 9\)
\(6x+70 = 11x - 9\)
\(70 + 9=11x - 6x\)
\(79 = 5x\)
\(x = 15.8\)

Step2: Solve for \(y\) in problem 5

Use the triangle - angle sum property (\(180^{\circ}\) in a triangle).
\((5y - 8)+30+(11x - 9)=180\)
Substitute \(x = 15.8\)
\((5y - 8)+30+(11\times15.8 - 9)=180\)
\((5y - 8)+30+(173.8 - 9)=180\)
\((5y - 8)+30 + 164.8=180\)
\(5y-8+194.8 = 180\)
\(5y+186.8 = 180\)
\(5y=180 - 186.8\)
\(5y=-6.8\)
\(y=-1.36\)

Step3: Solve for \(x\) in problem 6

Use the triangle - angle sum property (\(180^{\circ}\) in a triangle).
\((4x - 7)+(7y - 1)+86+(9x + 4)=180\)
\(13x+7y+82 = 180\)
\(13x+7y=98\)

Step4: Assume the triangle is isosceles (if we assume \((4x - 7)=(7y - 1)\))

\(4x-7y = 6\)

Step5: Solve the system of equations

We have the system \(

$$\begin{cases}13x + 7y=98\\4x-7y = 6\end{cases}$$

\)
Add the two equations: \((13x + 7y)+(4x-7y)=98 + 6\)
\(17x=104\)
\(x=\frac{104}{17}\approx6.12\)
Substitute \(x=\frac{104}{17}\) into \(4x-7y = 6\)
\(4\times\frac{104}{17}-7y=6\)
\(\frac{416}{17}-7y=6\)
\(-7y=6-\frac{416}{17}\)
\(-7y=\frac{102 - 416}{17}\)
\(-7y=\frac{-314}{17}\)
\(y=\frac{314}{119}\approx2.64\)

Answer:

For problem 5: \(x = 15.8,y=-1.36\)
For problem 6: \(x=\frac{104}{17},y=\frac{314}{119}\)