Chapter 4.2, Problem 49HP

To determine

To calculate:

Value of $y$ and $z$ in the figure at the right.

Answer to Problem 49HP

Value of $y$ and $z$ in the figure at the right is $13$ and $14$ respectively.

Explanation of Solution

Given information:

  

Calculation:

The measure of an exterior angle of a triangle is equal to the sum of the measure of the two remote interior angles.

  $\begin{array}{l}\left(5y+5\right)+\left(4z+9\right)=135\\ 5y+4z+14=135\\ 5y+4z+14-14=135-14\\ 5y+4z=121\end{array}$

Now, set $\left(4z+9\right)+\left(9y-2\right)$ equal to $180$

  $\begin{array}{l}\left(4z+9\right)+\left(9y-2\right)=1804z+9y+7=180\\ 4z+9y+7-7=180-7\\ 4z+9y=173\end{array}$

Thus, one obtains a system linear equations.

  $\begin{array}{l}5y+4z=\mathrm{121...}\left(1\right)\\ 4z+9y=\mathrm{173...}\left(2\right)\end{array}$

Subtract eq. $\left(1\right)$ from eq. $\left(2\right)$

  $4y=52$

Multiply each side of $4y=52$ by $\frac{1}{4}$

  $\begin{array}{l}y=52\left(\frac{1}{4}\right)\\ y=13\end{array}$

Now, substitute $y=13$ in eq. $\left(1\right)$

  $\begin{array}{l}5\left(13\right)+4z=121\\ 65+4z=121\end{array}$

Add to each side $65+4z=121$ by $-65$

  $\begin{array}{l}4z=121-65\\ 4z=56\end{array}$

Multiply ach side of $4z=56$ by $\frac{1}{4}$ .

  $\begin{array}{l}z=56\left(\frac{1}{4}\right)\\ z=14\end{array}$

Hence, $y=13,z=14$