### Graphs of Lines from Equations

The graph of any linear function can be plotted using the slope-intercept form of the equation.

**Step 1:** Solve the equation for \begin{align*}y\end{align*} if it is not already in the form \begin{align*}y=mx+b.\end{align*}

**Step 2:** To graph the function, start by plotting the \begin{align*}y\end{align*}-intercept.

**Step 3:** Use the slope to find another point on the line. From the \begin{align*}y\end{align*}-intercept, move to the right the number of units equal to the denominator of the slope and then up or down the number of units equal to the numerator of the slope. Plot the point.

**Step 4:** Connect these two points to form a line and extend the line.

Note that you can repeat Step 3 multiple times in order to find more points on the line if you wish.

Because the equations of horizontal and vertical lines are special, these types of lines can be graphed differently:

- The graph of a horizontal line will have an equation of the form \begin{align*}y=a\end{align*} where \begin{align*}a\end{align*} is the y-intercept of the line. You can simply draw a horizontal line through the y-intercept to sketch the graph.
- The graph of a vertical line will have an equation of the form \begin{align*}x=c\end{align*}, where \begin{align*}x\end{align*} is the x-intercept of the line. You can simply draw a vertical line through the x-intercept to sketch the graph.

#### Let's practice by finding the y-intercept and slope of the following function:

\begin{align*}4x-3y-9=0.\end{align*}

The first step is to rewrite the equation in the form \begin{align*}y=mx+b.\end{align*} To do this, solve the equation for ‘\begin{align*}y\end{align*}’.

\begin{align*}& \qquad 4x-3y-9=0 && \text{Subtract } x \text{ from both sides}\\ & 4x{\color{red}-4x}-3y-9=0{\color{red}-4x}\\ & \qquad \quad \text{-}3y-9=\text{-}4x && \text{Add 9 to both sides}\\ & \quad \ \ \text{-}3y-9{\color{red}+9}=\text{-}4x{\color{red}+9}\\ & \qquad \qquad \ \ \text{-}3y=\text{-}4x+9 && \text{Divide by -3}\\ & \qquad \qquad \quad \frac{\text{-}3y}{{\color{red}\text{-}3}}=\frac{\text{-}4x}{{\color{red}\text{-}3}}+\frac{9}{{\color{red}\text{-}3}}\\ & \qquad \qquad \quad \frac{\cancel{\text{-}3}y}{{\color{red}\cancel{\text{-}3}}}=\frac{\text{-}4x}{{\color{red}\text{-}3}}+\frac{9}{{\color{red}\text{-}3}}\\ & \qquad \qquad \qquad \boxed{y=\frac{4}{3}x-3}\end{align*}

The \begin{align*}y\end{align*}-intercept is (0, –3) and the slope is \begin{align*}\frac{4}{3}\end{align*}.

#### Now, let's graph the following function:

\begin{align*}y=\frac{\text{-}3}{5}x+7\end{align*}

The \begin{align*}y\end{align*}-intercept is (0, 7) and the slope is \begin{align*}\frac{\text{-}3}{5}\end{align*}. Begin by plotting the \begin{align*}y\end{align*}-intercept on the grid.

From the \begin{align*}y\end{align*}-intercept, move to the right (run) 5 units and then move downward (rise) 3 units. Plot a point here.

Join the points with a straight line. Use a straight edge to draw the line.

#### Finally, let's plot the following linear equations on a Cartesian grid:

- \begin{align*}x=\text{-}3\end{align*}

A line that has \begin{align*}x=\text{-}3\end{align*} as its equation passes through all points that have -3 as the \begin{align*}x-\end{align*}coordinate. The line also has a slope that is undefined. This line is parallel to the \begin{align*}y\end{align*}-axis.

- \begin{align*}y=5\end{align*}

A line that \begin{align*}y=5\end{align*} has as its equation passes through all points that have 5 as the \begin{align*}y-\end{align*}coordinate. The line also has a slope of zero. This line is parallel to the \begin{align*}x\end{align*}-axis.

### Examples

#### Example 1

Earlier, you were asked to plot the linear function \begin{align*}4y-5x=16\end{align*} on a Cartesian grid.

The first step is to rewrite the function in slope-intercept form.

\begin{align*}& \qquad 4y-5x=16 && \text{Add} \ 5x \ \text{to both sides of the equation.}\\ & 4y-5x{\color{red}+5x}=16{\color{red}+5x}\\ & \qquad \qquad \ 4y=16+5x && \text{Divide every term by} \ 4.\\ & \qquad \qquad \frac{4y}{{\color{red}4}}=\frac{16}{{\color{red}4}}+\frac{5x}{{\color{red}4}}\\ & \qquad \qquad \ \ y=4+\frac{5}{4}x && \text{Write the equation in the form} \ y=mx+b.\\ & \qquad \qquad \ \boxed{y=\frac{5}{4}x+4}\end{align*}

The slope of the line is \begin{align*}\frac{5}{4}\end{align*} and the \begin{align*}y\end{align*}-intercept is (0, 4)

Plot the \begin{align*}y\end{align*}-intercept at (0, 4). From the \begin{align*}y\end{align*}-intercept, move to the right 4 units and then move upward 5 units. Plot the point. Using a straight edge, join the points.

#### Example 2

Using the slope-intercept method, graph the linear function \begin{align*}y=\text{-}\frac{3}{2}x-1.\end{align*}

The slope of the line is \begin{align*}\text{-}\frac{3}{2}\end{align*} and the \begin{align*}y\end{align*}-intercept is (0, -1). Plot the \begin{align*}y\end{align*}-intercept. Apply the slope to the \begin{align*}y\end{align*}-intercept. Use a straight edge to join the two points.

#### Example 3

Using the slope-intercept method, graph the linear function \begin{align*}7x-3y-15=0.\end{align*}

Write the equation in slope-intercept form.

- \begin{align*}& \quad \ \ 7x-3y-15=0 && \text{Solve the equation for} \ y.\\ & 7x{\color{red}-7x}-3y-15=0{\color{red}-7x}\\ & \qquad \ \ \text{-}3y-15=\text{-}7x\\ & \quad \text{-}3y-15{\color{red}+15}=\text{-}7x{\color{red}+15}\\ & \qquad \qquad \quad \text{-}3y=\text{-}7x+15\\ & \qquad \qquad \quad \ \frac{\text{-}3y}{{\color{red}\text{-}3}}=\frac{\text{-}7x}{{\color{red}\text{-}3}}+\frac{15}{{\color{red}\text{-}3}}\\ & \qquad \qquad \qquad \ \boxed{y =\frac{7}{3}x-5}\end{align*}
- The slope is \begin{align*}\frac{7}{3}\end{align*} and the \begin{align*}y\end{align*}-intercept is (0, -5). Plot the \begin{align*}y\end{align*}-intercept. Apply the slope to the \begin{align*}y\end{align*}-intercept. Use a straight edge to join the two points.

#### Example 4

Graph the following lines on the same Cartesian grid. What shape is formed by the combination of the graphs a-d?

- \begin{align*}y=\text{-}3\end{align*}
- \begin{align*}x=4\end{align*}
- \begin{align*}y=2\end{align*}
- \begin{align*}x=-6\end{align*}

There are four lines to be graphed. The lines \begin{align*}a\end{align*} and \begin{align*}c\end{align*} are lines with a slope of zero and are parallel to the \begin{align*}x\end{align*}-axis. The lines \begin{align*}b\end{align*} and \begin{align*}d\end{align*} are lines that have a slope that is undefined and are parallel to the \begin{align*}x\end{align*}-axis. The shape formed by the intersections of the lines is a rectangle.

### Review

For each of the following linear functions, state the slope and the \begin{align*}y\end{align*}-intercept:

- \begin{align*}y=\frac{5}{8}x+3\end{align*}
- \begin{align*}4x+5y-3=0\end{align*}
- \begin{align*}4x-3y+21=0\end{align*}
- \begin{align*}y=-7\end{align*}
- \begin{align*}9y-8x=27\end{align*}

Using the slope-intercept method, graph the following linear functions:

- \begin{align*}3x+y=4\end{align*}
- \begin{align*}3x-2y=\text{-}4\end{align*}
- \begin{align*}2x+6y+18=0\end{align*}
- \begin{align*}3x+7y=0\end{align*}
- \begin{align*}4x-5y=\text{-}30\end{align*}
- \begin{align*}6x-2y=8\end{align*}

Graph the following linear equations and state the slope of the line:

- \begin{align*}x=\text{-}5\end{align*}
- \begin{align*}y=8\end{align*}
- \begin{align*}y=\text{-}4\end{align*}
- \begin{align*}x=7\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 4.4.