Equations of Lines: Slope, Intercept, and Applications (Chapter 7 Study Notes)

Graphs, Functions, and Applications

Finding Equations of Lines

This section introduces methods for finding the equation of a line in various scenarios, including when given the slope and y-intercept, the slope and a point, or two points. It also covers applications involving parallel and perpendicular lines, and solving real-world problems using linear functions.

• Slope-Intercept Form: The equation of a line with slope m and y-intercept b is given by: $y = mx + b$

• Point-Slope Form: The equation of a line with slope m passing through point $(x_1, y_1)$ is: $y - y_1 = m(x - x_1)$

• Parallel Lines: Lines that have the same slope but different y-intercepts.

• Perpendicular Lines: Lines whose slopes are negative reciprocals of each other.

Finding the Equation of a Line: Slope and Y-Intercept Given

When both the slope and y-intercept are known, use the slope-intercept form.

• Example: Find the equation of a line with slope $-3.2$ and y-intercept $(0, 5)$. Solution: Substitute $-3.2$$for$ and $5$for$b$:$y = mx + by = -3.2x + 5$ Application: This form is useful for quickly graphing a line or modeling linear relationships.

Finding the Equation of a Line: Slope and a Point Given

If the slope and a point on the line are given, use the point-slope form to find the equation.

• Point-Slope Equation: $y - y_1 = m(x - x_1)$

• Example: Find the equation of the line with slope $3$containing the point$(2, 7)$. Solution: $y - 7 = 3(x - 2)$ $y - 7 = 3x - 6$ $y = 3x + 1$ Alternate Method (Slope-Intercept): Substitute $x = 2$, $y = 7$, $m = 3$ into $y = mx + b$: $7 = 3(2) + b$ $7 = 6 + b$ $1 = b$ So, $y = 3x + 1$

Finding the Equation of a Line: Two Points Given

When two points are given, first calculate the slope, then use either the point-slope or slope-intercept form.

• Formula for Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Find the equation of the line containing points $(2, 2)$ and $(-6, -4)$. Solution: Calculate the slope: $m = \frac{-4 - 2}{-6 - 2} = \frac{-6}{-8} = \frac{3}{4}$ Use point-slope form with $(2, 2)$: $y - 2 = \frac{3}{4}(x - 2)$ $y - 2 = \frac{3}{4}x - \frac{6}{4}$ $y = \frac{3}{4}x - \frac{6}{4} + 2$ $y = \frac{3}{4}x + \frac{2}{4}$ $y = \frac{3}{4}x + \frac{1}{2}$

Parallel and Perpendicular Lines

Equations of lines parallel or perpendicular to a given line can be found by using the slope of the original line and a given point.

• Parallel Lines: Use the same slope as the given line. Example: Find the equation of the line containing point $(-5, 1)$ and parallel to $y = -3x + 4$. Solution: Slope $m = -3$ Use point-slope form: $y - 1 = -3(x + 5)$ $y - 1 = -3x - 15$ $y = -3x - 14$

• Perpendicular Lines: Use the negative reciprocal of the original slope. Example: Find the equation of the line containing point $(7, 1)$ and perpendicular to $7x - 2y = -2$. Solution: Rewrite $7x - 2y = -2$ in slope-intercept form: $-2y = -7x - 2$ $y = \frac{7}{2}x + 1$ Slope of perpendicular line: $m = -\frac{2}{7}$ Use point-slope form: $y - 1 = -\frac{2}{7}(x - 7)$ $y - 1 = -\frac{2}{7}x + 2$ $y = -\frac{2}{7}x + 3$

Applications: Linear Functions in Real-World Problems

Linear functions are used to model relationships in various contexts, such as cost, distance, and time.

• Example: Calvin Appliance charges a service fee and $55 per hour for labor. Formulate a linear function for the total cost$C(t)$, where$t$ is the number of hours. Solution: $C(t) = 55t + 35$ Graph: The y-intercept is $(0, 35)$, and the slope is $55$ (rate per hour).

• Application: To find the cost of a $2.5$ hour service call: $C(2.5) = 55(2.5) + 35$ $C(2.5) = 137.5 + 35 = 172.5$ The cost is $172.50$.

Summary Table: Forms of Linear Equations

Form	Equation	When to Use
Slope-Intercept	$y = mx + b$	When slope and y-intercept are known
Point-Slope	$y - y_1 = m(x - x_1)$	When slope and a point are known
Standard	$Ax + By = C$	General form; can be converted to other forms

Key Terms: Slope, y-intercept, point-slope form, parallel lines, perpendicular lines, linear function

Additional info: The notes are based on textbook slides for College Algebra, focusing on equations of lines and their applications in real-world contexts.