Equations of Lines and Linear Modeling in Precalculus

Section 1.4: Equations of Lines and Modeling

Introduction

This section covers the fundamental concepts of equations of lines, including how to determine equations for lines, analyze relationships between lines (parallelism and perpendicularity), and apply linear models to real-world data. These skills are essential for understanding linear functions and their applications in precalculus.

Equations of Lines

Slope-Intercept Equation

The slope-intercept form of a line is a widely used equation in algebra and precalculus. It is written as:

• General form: $y = mx + b$

• Where:

  • m = slope of the line

  • b = y-intercept (the value of y when x = 0)

If you know the slope and y-intercept, you can immediately write the equation of the line.

Example 1

• Given: Slope $m = -\frac{7}{9}$, y-intercept $(0, 16)$

• Equation: $y = -\frac{7}{9}x + 16$

Example 2

• Given: Slope $m = -\frac{2}{3}$, passes through $(-3, 6)$

• Substitute $x = -3$, $y = 6$ into $y = mx + b$ to solve for $b$:

• $6 = -\frac{2}{3}(-3) + b \implies 6 = 2 + b \implies b = 4$

• Equation: $y = -\frac{2}{3}x + 4$

Point-Slope Equation

The point-slope form is useful when you know the slope and a point $(x_1, y_1)$ on the line:

• Equation: $y - y_1 = m(x - x_1)$

Example 3

• Given: Points $(2, 3)$ and $(1, -4)$

• Find the slope: $m = \frac{-4 - 3}{1 - 2} = \frac{-7}{-1} = 7$

• Equation: $y - 3 = 7(x - 2)$, which simplifies to $y = 7x - 11$

Relationships Between Lines

Parallel Lines

Two lines are parallel if they have the same slope and different y-intercepts (unless they are the same line). Vertical lines (of the form $x = a$) are also parallel to each other.

• Condition: $m_1 = m_2$ and $b_1 \neq b_2$

Perpendicular Lines

Two lines are perpendicular if the product of their slopes is $-1$ (i.e., their slopes are negative reciprocals):

• Condition: $m_1 m_2 = -1$

• Additionally, a vertical line ($x = a$) and a horizontal line ($y = b$) are always perpendicular.

Example: Classifying Line Relationships

• Parallel: $y = 2x + 4$ and $y = 2x - 3$ (same slope, different intercepts)

• Perpendicular: $y = 3x + 1$ and $y = -\frac{1}{3}x + 4$ (slopes multiply to $-1$)

• Neither: $y = 2x + 1$ and $y = -3x + 4$ (slopes are not equal and not negative reciprocals)

Finding Equations of Parallel and Perpendicular Lines

Parallel Line Through a Point

• Given a line with slope $m$ and a point $(x_1, y_1)$, the parallel line has the same slope $m$.

• Use point-slope form: $y - y_1 = m(x - x_1)$

Perpendicular Line Through a Point

• Given a line with slope $m$, the perpendicular line has slope $-\frac{1}{m}$.

• Use point-slope form with the new slope and the given point.

Example

• Given: Line $4y - x = 20$ (rewrite as $y = \frac{1}{4}x + 5$), point $(2, -3)$

• Parallel line: Slope $\frac{1}{4}$, equation $y + 3 = \frac{1}{4}(x - 2)$

• Perpendicular line: Slope $-4$, equation $y + 3 = -4(x - 2)$

Mathematical Modeling and Linear Regression

Mathematical Models

A mathematical model uses mathematical language to describe real-world phenomena. Models allow us to make predictions and analyze situations. If a model does not fit experimental data, it must be revised or replaced.

Curve Fitting

Curve fitting is the process of finding a function that best fits a set of data points. When the data appears to follow a straight line, a linear function is used.

Example: Modeling Data with a Linear Function

• Given: Cost of a 30-second Super Bowl commercial increased from 2010 to 2015.

• Choose two data points, e.g., $(1, 3.1)$ and $(4, 4.0)$ (where $x$ is years after 2010, $y$ is cost in millions).

• Find the slope: $m = \frac{4.0 - 3.1}{4 - 1} = 0.3$

• Equation: $y - 3.1 = 0.3(x - 1)$, or $y = 0.3x + 2.8$

• Estimate for 2018 ($x = 8$): $y = 0.3(8) + 2.8 = 5.2$ (million dollars)

Linear Regression

Linear regression is a statistical method for finding the best-fitting straight line through a set of data points. The regression line has the form $y = mx + b$ and can be found using a graphing calculator or statistical software.

• Enter $x$-values (independent variable) and $y$-values (dependent variable) into lists.

• Use the calculator's linear regression feature to find $m$ and $b$.

• Example regression equation: $y = 0.3143x + 2.8476$

• Prediction for $x = 8$: $y = 0.3143(8) + 2.8476 \approx 5.3$

Correlation Coefficient

The correlation coefficient $r$ measures the strength and direction of a linear relationship between two variables. It ranges from $-1$ to $1$:

• $r = 1$: Perfect positive linear correlation (all points on a line with positive slope)

• $r = -1$: Perfect negative linear correlation (all points on a line with negative slope)

• $r = 0$: No linear correlation

• The closer $|r|$ is to 1, the stronger the linear relationship.

Value of r	Interpretation
1	Perfect positive correlation
0.9	High positive correlation
0.4	Low positive correlation
0	No correlation
-0.4	Low negative correlation
-0.9	High negative correlation
-1	Perfect negative correlation

Summary

• Equations of lines can be written in slope-intercept or point-slope form.

• Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Mathematical models and linear regression are used to fit and predict real-world data.

• The correlation coefficient quantifies the strength of a linear relationship between variables.