Cartesian Geometry Activity

Section 7.4 Cartesian Geometry Activity

Subsection 7.4.1 Equations of Lines

Activity 7.4.1.

Write the equation of the line with \(y\)-intercept \(7\) and slope \(\frac{-3}{2}\text{.}\)

Solution

We just need to put the slope and the intercept into the slope-intercept form of the equation of a line.

\begin{align*} y \amp = mx + b \\ y \amp = \frac{-3}{2} x + 7 \end{align*}

Activity 7.4.2.

Write the equation of the line with \(y\)-intercept \(-4\) and slope \(5\text{.}\)

Solution

We just need to put the slope and the intercept into the slope-intercept form of the equation of a line.

\begin{align*} y \amp = mx + b \\ y \amp = 5x - 4 \end{align*}

Activity 7.4.3.

Write the equation of the line with slope \(-2\) that goes through the point \((1,1)\text{.}\)

Solution

I start by putting the slope into into the slope-intercept form.

\begin{equation*} y = (-2) x + b \end{equation*}

I need to figure out the intercept. I can do this by putting the point into equation and solving for \(b\text{.}\)

\begin{align*} y \amp = -2x + b \\ 1 \amp = (-2)(1) + b \\ 1 \amp = -2 + b \\ 3 \amp = b \end{align*}

That gives me the value for the intercept, so I can finish the equation of the line.

\begin{equation*} y = -2x + 3 \end{equation*}

Activity 7.4.4.

Write the equation of the line with slope \(\frac{1}{4}\) that goes through the point \((-2,3)\text{.}\)

Solution

I start by putting the slope into into the slope-intercept form.

\begin{equation*} y = \frac{1}{4} x + b \end{equation*}

I need to figure out the intercept. I can do this by putting the point into equation and solving for \(b\text{.}\)

\begin{align*} y \amp = \frac{1}{4} x + b \\ 3 \amp = \frac{1}{4} (-2) + b \\ 3 \amp = \frac{-1}{2} + b \\ 3 + \frac{1}{2} \amp = b \\ \frac{7}{2} \amp = b \end{align*}

That gives me the value for the intercept, so I can finish the equation of the line.

\begin{equation*} y = \frac{1}{4} x + \frac{7}{2} \end{equation*}

Activity 7.4.5.

Write the equation of the line that goes through the point \((0,1)\) and \((5,5)\)

Solution

First I need to calculate the slope. I can do this by taking the difference of the \(y\) coordinates (the rise) and dividing by the difference of the \(x\) coordinates (the run).

\begin{equation*} m = \frac{5 - 1}{5 - 0} = \frac{5}{4} \end{equation*}

Now that I have the slope, I can put it into the slope-intercept form.

\begin{equation*} y = \frac{5}{4} x + b \end{equation*}

I need to figure out the intercept. I can do this by putting either point into equation and solving for \(b\text{.}\)

\begin{align*} y \amp = \frac{5}{4}x + b \\ 5 \amp = \frac{5}{4} (5) + b \\ 5 \amp = \frac{25}{4} + b \\ 5 - \frac{25}{4} \amp = b \\ \frac{-5}{4} \amp = b \end{align*}

That gives me the value for the intercept, so I can finish the equation of the line.

\begin{equation*} y = \frac{5}{4} x - \frac{5}{4} \end{equation*}

Activity 7.4.6.

Write the equation of the line that goes through the point \((-2,-2)\) and \((6,1)\)

Solution

First I need to calculate the slope. I can do this by taking the difference of the \(y\) coordinates (the rise) and dividing by the difference of the \(x\) coordinates (the run).

\begin{equation*} m = \frac{1 - (-2)}{6 - (-2)} = \frac{3}{8} \end{equation*}

Now that I have the slope, I can put it into the slope-intercept form.

\begin{equation*} y = \frac{3}{8} x + b \end{equation*}

I need to figure out the intercept. I can do this by putting either point into equation and solving for \(b\text{.}\)

\begin{align*} y \amp = \frac{3}{8}x + b \\ -2 \amp = \frac{3}{8} (-2) + b \\ -2 \amp = \frac{-3}{4} + b \\ -2 + \frac{3}{4} \amp = b \\ \frac{-5}{4} \amp = b \end{align*}

That gives me the value for the intercept, so I can finish the equation of the line.

\begin{equation*} y = \frac{3}{8} x - \frac{5}{4} \end{equation*}