Problem 23 - A Graphical Approach to College Algebra

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. $$\begin{array}{c|c}x & y \\\\\hline-7 & -44 \\\\-6 & -36 \\\\-5 & -28 \\\\-4 & -20\end{array}$$

Step-by-Step Solution

Answer

The equation is \( y = 8x + 12 \).

Step 1: Understand the Slope-Intercept Formula

The slope-intercept form of a line is written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Our goal is to find both \( m \) and \( b \).

Step 2: Determine the Slope \( m \)

The slope \( m \) can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We'll use the points \((-7, -44)\) and \((-6, -36)\) for this computation. Substituting in these coordinates, we get: \[ m = \frac{-36 - (-44)}{-6 - (-7)} = \frac{8}{1} = 8 \] So, the slope \( m \) is 8.

Step 3: Identify the Y-Intercept \( b \)

Now that we have the slope, we need to find the y-intercept \( b \). We can use one of the given points and the value for \( m \) in the slope-intercept formula. Using the point \((-6, -36)\), substitute into the equation \( y = mx + b \): \[ -36 = 8(-6) + b \] \[ -36 = -48 + b \] \[ b = -36 + 48 \] \[ b = 12 \] Therefore, the y-intercept \( b \) is 12.

Step 4: Write the Equation in Slope-Intercept Form

With the slope \( m = 8 \) and the y-intercept \( b = 12 \), substitute these values into the slope-intercept form: \[ y = 8x + 12 \] So, the equation of the line in slope-intercept form is \( y = 8x + 12 \).

Key Concepts

Slope

The slope of a line is a measure of its steepness, commonly denoted by the letter \( m \). When two points on a line are given, the slope can be found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula calculates the change in the y-coordinates (vertical change) divided by the change in the x-coordinates (horizontal change).
For example, using points \((-7, -44)\) and \((-6, -36)\), we calculate the slope as follows:

• Subtract the y-values: \(-36 - (-44) = 8\)
• Subtract the x-values: \(-6 - (-7) = 1\)
• Now divide the results: \( \frac{8}{1} = 8 \)

This means the slope \( m \) is 8. A positive slope indicates the line ascends from left to right. Conversely, a negative slope would show a descending line.

y-intercept

To find the y-intercept, represented by \( b \), we take the value of \( y \) when \( x \) is 0. In the slope-intercept form \( y = mx + b \), \( b \) is where the line crosses the y-axis. To determine \( b \), use a known point and the slope:

• Choose a point, such as \((-6, -36)\).
• Substitute into the equation: \(-36 = 8(-6) + b\).
• Solve for \( b \): find \(-36 + 48 = 12\).

Thus, the y-intercept \( b \) is 12. This point \((0, 12)\) is crucial as it provides a starting point for graphing the linear equation.

Linear Equations

Linear equations form straight lines on a graph, and the slope-intercept form, \( y = mx + b \), is one of several ways to write them. In this format:

• The slope \( m \) dictates the incline or decline.
• The y-intercept \( b \) specifies where the line intersects the y-axis.
• Both components are essential for defining the line's direction and position.

Linear equations provide a predictable relationship between \( x \) and \( y \). Any increase/decrease in \( x \) leads to a proportional increase/decrease in \( y \). For example, in the equation \( y = 8x + 12 \), every unit increase in \( x \) results in an 8-unit increase in \( y \). This predictable nature makes linear equations a fundamental concept in mathematics and various applications.