Question

layla borrowed some money from her friend in order to help buy a new video game system. layla agreed to pay the friend back some amount every week until her loan was paid off. let ( l ) represent the amount layla owes her friend after ( t ) weeks. the table below has select values showing the linear relationship between ( t ) and ( l ). determine how many weeks when the amount of money layla owed her friend was $120. \\(\

$$\begin{array}{|c|c|} \\hline t & l \\\\ \\hline 1 & 160 \\\\ \\hline 7 & 100 \\\\ \\hline 11 & 60 \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Find the slope (rate of change)

The slope \( m \) between two points \((t_1, L_1)\) and \((t_2, L_2)\) is given by \( m=\frac{L_2 - L_1}{t_2 - t_1} \). Using the points \((1, 160)\) and \((7, 100)\):
\( m=\frac{100 - 160}{7 - 1}=\frac{- 60}{6}=- 10 \)

Step2: Write the linear equation

Using the point - slope form \( L - L_1=m(t - t_1) \) with the point \((1, 160)\) and \( m=-10 \):
\( L - 160=-10(t - 1) \)
Simplify the equation:
\( L - 160=-10t + 10 \)
\( L=-10t + 170 \)

Step3: Solve for \( t \) when \( L = 120 \)

Substitute \( L = 120 \) into the equation \( L=-10t + 170 \):
\( 120=-10t + 170 \)
Subtract 170 from both sides:
\( 120-170=-10t \)
\( - 50=-10t \)
Divide both sides by - 10:
\( t = 5 \)

Answer:

5