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:

\(y=-25 x+160\)