Question

sections 2 - 1b and 2 - 2: slope - intercept and point - slope form

1. at eagle bay, it costs $12 per hour to rent a canoe. nate and his friends rented a canoe for 4 hours and paid $68. write a linear equation in slope - intercept form to model the cost (y) of renting a canoe for (x) number of hours.
2. jeff is keeping track of his weight over several weeks. after 2 weeks, he weighs 194 pounds. after 6 weeks, he weighs 186 pounds. write a linear equation in point - slope form to model the number of weeks (x) and jeffs weight (y).

section 2 - 3: standard form

1. write a linear equation in standard form of a line with a slope of $\frac{5}{6}$ and y - intercept = 2.
2. write a linear equation in standard form of a line with a slope of $-\frac{1}{4}$ and y - intercept = - 8.

Explanation:

Step1: Find the y - intercept for the canoe - renting problem

The slope - intercept form is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. The cost per hour (slope $m$) is $12$. They rented for $x = 4$ hours and paid $y=68$. Substitute into $y = mx + b$: $68=12\times4 + b$.
$68 = 48 + b$, so $b=68 - 48=20$. The equation is $y = 12x+20$.

Step2: Find the slope for Jeff's weight problem

The point - slope form is $y - y_1=m(x - x_1)$. We have two points $(x_1,y_1)=(2,194)$ and $(x_2,y_2)=(6,186)$. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{186 - 194}{6 - 2}=\frac{- 8}{4}=-2$. Using the point $(2,194)$, the point - slope form is $y - 194=-2(x - 2)$.

Step3: Write the standard form for the first standard - form problem

The slope - intercept form is $y=\frac{5}{6}x + 2$. Multiply through by 6 to get $6y = 5x+12$. Rearrange to standard form $Ax + By=C$: $5x-6y=-12$.

Step4: Write the standard form for the second standard - form problem

The slope - intercept form is $y=-\frac{1}{4}x - 8$. Multiply through by 4 to get $4y=-x - 32$. Rearrange to standard form: $x + 4y=-32$.

Answer:

1. $y = 12x + 20$
2. $y - 194=-2(x - 2)$
3. $5x-6y=-12$
4. $x + 4y=-32$