Writing Slope-Intercept Equations

1. $y=0.1 x+100$

The fee for every vertical meter climbed is constant, so we are dealing with a linear relationship.

Let's interpret the meaning of the given information in terms of the line representing this relationship.

The initial fee is $\$100$. This corresponds to the point $(0, 100)$, which is also the $y$-intercept.

The total fee for climbing up $3000$ meters is $$400$, which corresponds to the point $(3000,400)$.

Let's use the slope formula with the points $(0,100)$ and $3000,400)$.

$m=\frac{400-100}{3000-0}=\frac{300}{3000}=0.1$

This means that the agency charges a constant fee of $$0.1$ per vertical meter climbed.

Now we know the slope of the line is $0.1$ and the $y$-intercept is $(0,100)$, so we can write the equation of that line:

$y=0.1 x+100$

2. $y=-24 x+166$

Rachel drove at a constant rate, so we are dealing with a linear relationship.

Let's interpret the meaning of the given information in terms of the line representing this relationship.

Rachel drove $24$ meters per second. This corresponds to a slope with an absolute value of $24$.

Notice that Rachel is driving closer to the safe zone. So our line is decreasing, which means the slope is $-24$.

After $4$ seconds of driving, she was $70$ meters away from the safe zone. This corresponds to the point $(4, 70)$.

So the slope of the relationship's line is $-24$ and the line passes through $(4, 70)$.

Let's find the $y$-intercept, represented by the point $(0, b)$, using the slope formula:

$\frac{b-70}{0-4}=-24$

Solving this equation, we get $b=166$.

Show me the solution.

$\begin{aligned} \frac{b-70}{0-4} &=-24 \\ b-70 &=-24(-4) \\ b-70 &=96 \\ b &=166 \end{aligned}$

Now we know the slope of the line is $-24$ and the $y$-intercept is $(0, 166)$, so we can write the equation of that line:

$y=-24 x+166$

3. $y=6 x+8$

Carolina's hourly fee is constant, so we are dealing with a linear relationship.

Let's interpret the meaning of the given information in terms of the line representing this relationship.

Carolina's fee increases at a rate of $$6$ per hour. This corresponds to a slope of $6$.

Carolina's total fee for a $4$-hour job is $$32$. This corresponds to the point $(4,32)$.

So the slope of the relationship's line is $6$ and the line passes through $(4, 32)$.

Let's find the $y$-intercept, represented by the point $(0,b)$, using the slope formula:

$\frac{b-32}{0-4}=6$

Solving this equation, we get $b =8$.

Show me the solution.

$\begin{aligned} \frac{b-32}{0-4} &=6 \\ b-32 &=6(-4) \\ b-32 &=-24 \\ b &=8 \end{aligned}$

Now we know the slope of the line is $6$ and the $y$-intercept is $(0,8)$, so we can write the equation of that line:

$y=6 x+8$

4. $y=-25 x+160$

Kayden drove at a constant rate, so we are dealing with a linear relationship.

Let's interpret the meaning of the given information in terms of the line representing this relationship.

The initial distance to drive was $160$ meters. This corresponds to the point $(0, 160)$, which is also the $y$-intercept.

There were $85$ meters left after $3$ seconds, which corresponds to the point $(3,85)$.

Let's use the slope formula with the points $(0,160)$ and $(3,85)$.

$m=\frac{85-160}{3-0}=\frac{-75}{3}=-25$

This means that the distance to the safe zone decreased by $25$ meters per second (because Kayden drove at a speed of $25$ meters per second).

Now we know the slope of the line is \greenD{-25}−25start color #1fab54, minus, 25, end color #1fab54 and the yyy-intercept is $-25$ and the $y$-intercept is $(0, 160)$, so we can write the equation of that line:

$y=-25 x+160$