Question

For the following figure, $\mathbf{\text{E}}$is the midpoint of $\overline{\mathbf{D}\mathbf{F}}$. Find the value $\mathit{y}$, where $\mathit{G}\mathit{E}\mathbf{=}\mathbf{3}\mathit{y}$, $\mathit{G}\mathit{H}\mathbf{=}\mathbf{7}\mathit{y}\mathbf{-}\mathbf{4}$and $\mathit{E}\mathit{H}\mathbf{=}\mathbf{24}$

Short answer

The value of$y$ is 7.

Step-by-step solution

Step 1. Apply the concept of the length of the line.

The total length of the line is the sum of the length of all parts of a line.

Step 2. Check the figure.

Consider the figure.

Here the length of $GH$ is the sum of$GE$ and $EH$

Step 3. Find the value of  y.

Since $GH=GE+EH$

Equate the values as follows:

$7y-4=3y+24$

Step 4. Simplify the equation.

Simplify the equation as follows:

$\begin{array}{c}7y-4=3y+24\\ 7y-3y=24+4\\ 4y=28\end{array}$

Divide the equation both sides by 4.

$\begin{array}{c}\frac{4y}{4}=\frac{28}{4}\\ y=7\end{array}$

Therefore, the value of$y$ is 7.