Writing Slope-Intercept Equations

1. Find the equation of the line.
Use exact numbers.

$y = \text { _____ }x + \text { _____ }$

2. Find the equation of the line.
Use exact numbers.

$y = \text { _____ }x + \text { _____ }$

3. Find the equation of the line.

Use exact numbers.

$y = \text { _____ }x + \text { _____ }$

4. Find the equation of the line.

Use exact numbers.

$y = \text { _____ }x + \text { _____ }$

1. $y=-\frac{3}{2} x+3$.

We are asked to complete the equation in slope-intercept form: $y=m \cdot x+b$. In this form, $m$ gives us the slope of the line and $b$ gives us its $y$-intercept.

By looking at the graph, we can see that the $y$-intercept is $(0,3)$.

In order to find the slope, we need another point on the line whose coordinates are clearly visible. Such a point is $(2,0)$.

Now, to find the slope, we need to take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{0-3}{2-0}=\frac{-3}{2}=-\frac{3}{2}$

The equation is $y=-\frac{3}{2} x+3$.

2. $y=2 x+4$.

We are asked to complete the equation in slope-intercept form: $y=m \cdot x+b$. In this form, $m$ gives us the slope of the line and $b$ gives us its $y$-intercept.

By looking at the graph, we can see that the $y$-intercept is $(0,4)$.

In order to find the slope, we need another point on the line whose coordinates are clearly visible. Such a point is $(1,6)$.

Now, to find the slope, we need to take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{6-4}{1-0}=\frac{2}{1}=2$

The equation is $y=2 x+4$.

3. $y=\frac{3}{4} x-2$.

We are asked to complete the equation in slope-intercept form: $y=m \cdot x+b$. In this form, $m$ gives us the slope of the line and $b$ gives us its $y$-intercept.

By looking at the graph, we can see that the $y$-intercept is $(4,1)$.

Now, to find the slope, we need to take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{1-(-2)}{4-0}=\frac{3}{4}$

The equation is $y=\frac{3}{4} x-2$.

4. $y=x-5$.

We are asked to complete the equation in slope-intercept form: $y=m \cdot x+b$. In this form, $m$ gives us the slope of the line and $b$ gives us its $y$-intercept.

By looking at the graph, we can see that the $y$-intercept is $(1,-4)$.

In order to find the slope, we need another point on the line whose coordinates are clearly visible. Such a point is $(1,-4)$.

Now, to find the slope, we need to take the ratio of the corresponding differences in the $y$-values and the $x$-values:

$\frac{-4-(-5)}{1-0}=\frac{1}{1}=1$

The equation is $y=x-5$.

1. Complete the equation of the line through $(2,1)$ and $(5,-8)$.
Use exact numbers.

$y = \text { ______ }$

2. Complete the equation of the line through $(3, -8)$ and $(6, -4)$.
Use exact numbers.

$y = \text { ______ }$

3. Complete the equation of the line through $(3, -1)$ and $(4, 7)$.
Use exact numbers.

$y = \text { ______ }$

4. Complete the equation of the line through $(-10, -7)$ and $(-5, -9)$.
Use exact numbers.

$y = \text { ______ }$

1.$y=-3 x+7$

Let's find the slope:

$\begin{aligned} \text { Slope } &=\frac{(-8)-1}{5-2} \\ &=\frac{-9}{3} \\ &=-3 \end{aligned}$

The equation is $y=-3 x+b$ for some $b$.

Let's plug the point $(2,1)$ to find $b$:

$\begin{aligned} &y=-3 x+b \\ &1=-3(2)+b \\ &1=-6+b \\ &7=b \end{aligned}$

The equation is $y=-3 x+7$.

2. $y=\frac{4}{3} x-12$

Let's find the slope:

$\begin{aligned} \text { Slope } &=\frac{-4-(-8)}{6-3} \\ &=\frac{4}{3} \end{aligned}$

The equation is $y=\frac{4}{3} x+b$ for some $b$.

Let's plug the point $(6,−4)$to find $b$:

$\begin{aligned} y &=\frac{4}{3} x+b \\ -4 &=\frac{4}{3}(6)+b \\ -4 &=8+b \\ -12 &=b \end{aligned}$

The equation is $y=\frac{4}{3} x-12$.

3. $y=8 x-25$.

Let's find the slope:

$\begin{aligned} \text { Slope } &=\frac{7-(-1)}{4-3} \\ &=\frac{8}{1} \\ &=8 \end{aligned}$

The equation is $y=8 x+b$ for some $b$.

Let's plug the point $(4, 7)$ to find $b$:

$\begin{aligned} y &=8 x+b \\ 7 &=8(4)+b \\ 7 &=32+b \\ -25 &=b \end{aligned}$

The equation is $y=8 x-25$.

4. $y=-\frac{2}{5} x-11$

Let's find the slope:

$\begin{aligned} \text { Slope } &=\frac{-9-(-7)}{-5-(-10)} \\ &=\frac{-2}{5} \\ &=-\frac{2}{5} \end{aligned}$

The equation is $y=-\frac{2}{5} x+b$ for some $b$.

Let's plug the point $(−5,−9)$ to find $b$:

$\begin{aligned} y &=-\frac{2}{5} x+b \\ -9 &=-\frac{2}{5}(-5)+b \\ -9 &=2+b \\ -11 &=b \end{aligned}$

The equation is $y=-\frac{2}{5} x-11$.

1. Simba Travel Agency arranges trips for climbing Mount Kilimanjaro. For each trip, they charge an initial fee of $$100$ in addition to a constant fee for each vertical meter climbed. For instance, the total fee for climbing to the Shira Volcanic Cone, which is $3000$ meters above the base of the mountain, is $$400$.

Let $y$ represent the total fee (in dollars) of a trip where they climbed $x$ vertical meters.

Complete the equation for the relationship between the total fee and vertical distance.

$y = \text { ______ }$

2. Rachel is a stunt driver. One time, during a gig where she escaped from a building about to explode(!), she drove to get to the safe zone at $24$ meters per second. After $4$ seconds of driving, she was $70$ meters away from the safe zone.

Let $y$ represent the distance (in meters) from the safe zone after $x$ seconds.

Complete the equation for the relationship between the distance and number of seconds.

$y = \text { ______ }$

3. Carolina is mowing lawns for a summer job. For every mowing job, she charges an initial fee plus $$6$ for each hour of work. Her total fee for a $4$-hour job, for instance, is $$32$.

Let $y$ represent Carolina's fee (in dollars) for a single job that took $x$ hours for her to complete.

Complete the equation for the relationship between the fee and number of hours.

$y = \text { ______ }$

4. Kayden is a stunt driver. One time, during a gig where she escaped from a building about to explode(!), she drove at a constant speed to get to the safe zone that was $160$ meters away. After $3$ seconds of driving, she was $85$ meters away from the safe zone.

Let $y$ represent the distance (in meters) from the safe zone after $x$ seconds.

Complete the equation for the relationship between the distance and number of seconds.

$y = \text { ______ }$

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:

$y=-25 x+160$