Question

1. the cost for 7 dance lessons is $82. the cost for 11 dance lessons is $122. write a linear equation to find the total cost c for ℓ lessons. then use the equation to find the cost of 4 lessons.

1. the cost for 5 music lessons is $95. the cost for 10 music lessons is $165. write a linear equation to find the total cost c for n lessons. then use the equation to find the cost of 8 lessons.

1. it is 76°f at the 6000 - foot level of a mountain, and 49°f at the 12,000 - foot level of the mountain. write a linear equation to find the temperature t at an elevation x on the mountain, where x is in thousands of feet.

Explanation:

Problem 6

Step1: Find the slope (rate of change)

We have two points: \((\ell_1, C_1) = (7, 82)\) and \((\ell_2, C_2) = (11, 122)\). The slope \(m\) is calculated as \(m=\frac{C_2 - C_1}{\ell_2 - \ell_1}=\frac{122 - 82}{11 - 7}=\frac{40}{4} = 10\).

Step2: Use point - slope form to find the equation

Using the point - slope form \(C - C_1=m(\ell-\ell_1)\) with \((\ell_1, C_1)=(7,82)\) and \(m = 10\).
\(C-82 = 10(\ell - 7)\)
Expand the right - hand side: \(C-82=10\ell-70\)
Add 82 to both sides: \(C = 10\ell+12\)

Step3: Find the cost of 4 lessons

Substitute \(\ell = 4\) into the equation \(C = 10\ell+12\).
\(C=10\times4 + 12=40 + 12=52\)

Step1: Find the slope (rate of change)

We have two points: \((n_1, C_1)=(5,95)\) and \((n_2, C_2)=(10,165)\). The slope \(m\) is calculated as \(m=\frac{C_2 - C_1}{n_2 - n_1}=\frac{165 - 95}{10 - 5}=\frac{70}{5}=14\).

Step2: Use point - slope form to find the equation

Using the point - slope form \(C - C_1=m(n - n_1)\) with \((n_1, C_1)=(5,95)\) and \(m = 14\).
\(C - 95=14(n - 5)\)
Expand the right - hand side: \(C-95 = 14n-70\)
Add 95 to both sides: \(C=14n + 25\)

Step3: Find the cost of 8 lessons

Substitute \(n = 8\) into the equation \(C = 14n+25\).
\(C=14\times8+25=112 + 25=137\)

Step1: Find the slope (rate of change)

We have two points: \((X_1, T_1)=(6,76)\) (since 6000 feet is 6 thousand feet) and \((X_2, T_2)=(12,49)\). The slope \(m\) is calculated as \(m=\frac{T_2 - T_1}{X_2 - X_1}=\frac{49 - 76}{12 - 6}=\frac{- 27}{6}=-4.5\)

Step2: Use point - slope form to find the equation

Using the point - slope form \(T - T_1=m(X - X_1)\) with \((X_1, T_1)=(6,76)\) and \(m=-4.5\).
\(T - 76=-4.5(X - 6)\)
Expand the right - hand side: \(T-76=-4.5X + 27\)
Add 76 to both sides: \(T=-4.5X+103\)

Answer:

The linear equation is \(C = 10\ell+12\), and the cost of 4 lessons is \(\$52\).

Problem 7