Linear Equations, Parallel and Perpendicular Lines, and Linear Modeling 2.3

Linear Equations and Slope-Intercept Form

Definition and Formulation

Linear equations are fundamental in algebra and describe straight lines on the Cartesian plane. The most common form is the slope-intercept form:

• Slope-intercept form: $y = mx + b$, where m is the slope and b is the y-intercept.

• Slope (m): Measures the steepness of the line, calculated as the change in y over the change in x.

• Y-intercept (b): The point where the line crosses the y-axis.

Example: The equation $y = -\frac{2}{3}x + 2$ is in slope-intercept form, with slope $-\frac{2}{3}$ and y-intercept $2$.

Parallel and Perpendicular Lines

Parallel Lines

Parallel lines have the same slope but different y-intercepts. They never intersect.

• Condition: Two lines are parallel if their slopes are equal.

• Example: The line parallel to $2x + 3y = 6$ has the same slope as the original line.

• Convert $2x + 3y = 6$ to slope-intercept form:

$\begin{align*} 2x + 3y &= 6 \\ 3y &= -2x + 6 \\ y &= -\frac{2}{3}x + 2 \end{align*}$

• Slope is $-\frac{2}{3}$.

• Equation of a parallel line passing through $(3,6)$:

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

$\begin{align*} y - 6 &= -\frac{2}{3}(x - 3) \\ y - 6 &= -\frac{2}{3}x + 2 \\ y &= -\frac{2}{3}x + 8 \end{align*}$

Additional info: The final equation is in slope-intercept form and passes through the given point.

Perpendicular Lines

Perpendicular lines intersect at a right angle. Their slopes are negative reciprocals.

• Condition: If a line has slope $m$, a perpendicular line has slope $-\frac{1}{m}$.

• Example: The line perpendicular to $2x + 3y = 6$ has slope $\frac{3}{2}$.

• Equation of a perpendicular line passing through $(3,6)$:

Use point-slope form: $y - 6 = \frac{3}{2}(x - 3)$

$\begin{align*} y - 6 &= \frac{3}{2}(x - 3) \\ y - 6 &= \frac{3}{2}x - \frac{9}{2} \\ y &= \frac{3}{2}x - \frac{9}{2} + 6 \\ y &= \frac{3}{2}x + \frac{3}{2} \end{align*}$

Additional info: The negative reciprocal property ensures perpendicularity.

Point-Slope Form

Definition and Application

The point-slope form is useful for writing the equation of a line when a point and the slope are known:

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

• Convert to slope-intercept form by solving for $y$.

Example: For a line with slope $m = -3$ passing through $(3,6)$:

$\begin{align*} y - 6 &= -3(x - 3) \\ y - 6 &= -3x + 9 \\ y &= -3x + 15 \end{align*}$

Linear Modeling with Data

Fitting a Linear Model

When data changes at a fairly constant rate, a linear model can be used to describe the relationship. The rate of change is the slope of the line.

• Given data: Year and corresponding value (e.g., cost).

• Calculate slope $m$ using two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Example: Using the data points (2015, 2955) and (2021, 3501):

$\begin{align*} m &= \frac{3501 - 2955}{6 - 0} \\ m &= \frac{546}{6} \\ m &= 91 \end{align*}$

• Linear model: $y = 91x + 2955$

Using the Model for Prediction

Once the linear equation is established, it can be used to predict future values.

• Example: Predict the value for 2022 (7 years after 2015):

$\begin{align*} y &= 91(7) + 2955 \\ y &= 637 + 2955 \\ y &= 3592 \end{align*}$

Tabular Data: Year vs. Value

Purpose: Organizing Data for Linear Modeling

Year	Value (f)
2015	2955
2016	3038
2017	3156
2018	3242
2019	3312
2020	3377
2021	3501

Additional info: The table is used to visualize the trend and calculate the slope for the linear model.

Summary of Key Concepts

• Slope-intercept form is used to easily identify the slope and y-intercept of a line.

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

• Point-slope form is useful for writing equations when a point and slope are known.

• Linear models can be fitted to data that changes at a constant rate, allowing for predictions.