Question

question enola borrowed some money from her friend in order to help buy a new video game system. enola agreed to pay the friend back some amount every week until her loan was paid off. let l represent the amount enola owes her friend after t weeks. a graph of l is shown below. write an equation for l then state the x - intercept of the graph and determine its interpretation in the context of the problem.

Explanation:

Step1: Find the slope of the line

The line passes through points $(0, 100)$ and $(5,0)$. The slope $m$ of a line passing through $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1 = 0,y_1=100,x_2 = 5,y_2 = 0$. So $m=\frac{0 - 100}{5-0}=\frac{- 100}{5}=-20$.

Step2: Find the y - intercept

The y - intercept $b$ is the value of $L$ when $t = 0$. From the graph, when $t = 0$, $L=100$, so $b = 100$.

Step3: Write the equation of the line

The equation of a line in slope - intercept form is $L=mt + b$. Substituting $m=-20$ and $b = 100$, we get $L=-20t + 100$.

Step4: Find the x - intercept

The x - intercept is the value of $t$ when $L = 0$. Set $L = 0$ in the equation $L=-20t + 100$. Then $0=-20t+100$. Solving for $t$:
\[

$$\begin{align*} 20t&=100\\ t& = 5 \end{align*}$$

\]

Step5: Interpret the x - intercept

In the context of the problem, the x - intercept $(5,0)$ means that after 5 weeks, Enola will have paid off her entire loan (the amount she owes her friend is $0$).

Answer:

The equation for $L$ is $L=-20t + 100$. The x - intercept is $t = 5$, which means it takes 5 weeks for Enola to pay off her loan.