Sections 2.4 & 2.5 - Lines and Linear Equations in Two Variables

Section Objectives

1. Find and apply the slope-intercept form of the equation of a line.
2. 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 lines 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 $y=5x-8$.