# Sections 2.3 - Equations of Lines in Special Forms

Section Objectives

1. Find and apply the slope-intercept form of the equation of a line.
2. Find and apply the point-slope form of the equation of a line.
3. Graph a line using its slope and a point.
4. Find lines parallel or perpendicular to given lines.
5. Apply lines and linear equations in real-world applications.

### Point-Slope Form

Suppose a line with slope $m$ passes through the point $(x_0,y_0)$. Then for any other point $(x,y)$ on the line, it must be true that

$\displaystyle \frac{y-y_0}{x-x_0}=m$ or $y-y_0=m(x-x_0)$.

If we think of $(x,y)$ as a variable point on the line, then these expressions give us equations of the line. The equation in the form $y-y_0=m(x-x_0)$ is called point-slope form.

#### Examples

• Find an equation of the line with slope $-5$ passing through the point $(2,3)$.

• Find an equation of the line passing through the points $(1,4)$ and $(-3,7)$. Write your final answer in standard form.

• Find an equation of the line with $y$-intercept $(0,-4)$ and slope $2/3$.

### Slope-Intercept Form

In the last example, you may have noticed something interesting. When a linear equation is written in the form $y=mx+b$, we can immediately read off the slope and $y$-intercept. An equation in the form $y=mx+b$ is called slope-intercept form.

#### Examples

• Rewrite the standard form equation $4x-2y=14$ in slope-intercept form. Then determine the slope and $y$-intercept of the line described by the equation.

• A line with $y$-intercept $(0,2)$ passes through the other point $(1,9)$. Find the point-slope form of the equation for the line. Also write the equation is standard form.

• A line passes through the point $(2,-5)$ and is perpendicular to the line described by $y=2x-4$. Find an equation for the line.

• Don't forget about horizontal and vertical lines.... Find an equation of the horizontal line that passes through the $y$-intercept of the line described by $y=5x-8$.

• Determine an equation that converts temperatures in degrees Fahrenheit to degrees Celsius.

What if we wanted a formula that converts in the other direction?

• An engineer working with a certain voltage-controlled amplifier finds that an input of 1.32 volts is amplified to 14.00 volts, while an input of 3.79 volts has an output of 8.00 volts. Assuming the gain is linear, find an equation relating the input and output voltages.