Problem 63 - College Algebra Essentials

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. $$8 x-4 y-12=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = 2x - 3\). The slope is 2 and the y-intercept is -3. The graph of the equation would be a straight line crossing the y-axis at -3, with a steepness or slope of 2.

Step 1: Convert to Slope-Intercept Form

First, rewrite the provided equation, \(8x - 4y - 12 = 0\), to slope-intercept form. This can be done by isolating y. Add 4y to both sides and then subtract 12 from both sides to isolate y on one side of the equation. Divide everything by 4, the coefficient in front of y, to get the equation into the correct form. The equation in slope-form is: \(y = 2x - 3\).

Step 2: Identify Slope and Y-Intercept

From the rewritten equation, identify the slope (m) and y-intercept (b). In our equation, \(y = 2x - 3\), the coefficient in front of x (2) is the slope (m), and the constant term (-3) is the y-intercept (b). Therefore, slope (m) = 2, and the y-intercept (b) = -3.

Step 3: Graph the Linear Equation

To graph this equation, start by labeling the y-intercept. From Step 2, the intercept is -3, so place a point on the y-axis at -3. Then, from that point, use the slope to find the next point. The slope is 2, meaning there is a rise of 2 for a run of 1. Thus, move up two places and over one place to the right on the graph. Connect the points with a line and extend it in both directions, representing all possible solutions for the equation.

Key Concepts

Linear Equations

A linear equation is any equation that graphs a straight line when we plot it on a coordinate plane. It represents a constant rate of change and can be written in various forms. The most common is the slope-intercept form, which is expressed as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.

Our task is to convert linear equations into this form because it provides clear information for graphing.

• The slope \(m\) tells us how steep the line is.
• The y-intercept \(b\) indicates where the line crosses the y-axis.

Understanding these components makes it easier to identify the behavior and positioning of the line within a graph.

Graphing

Graphing a linear equation involves plotting its y-intercept and then using the slope to determine the line's direction and steepness. For the equation \(y = 2x - 3\):

• Start by addressing the y-intercept, which is \(-3\). Place a point at \( (0, -3) \) on the y-axis.
• Use the slope \(2\) or \(\frac{2}{1}\) to guide the plotting of the next point. From the y-intercept, rise 2 units and run 1 unit to the right to place another point.
• Connect these points with a straight line, extending it evenly in both directions.

The line visually represents all solutions to the equation. Graphing demonstrates the relationship between the variables and how changes in \(x\) result in changes in \(y\).

Remember, each point on the line is a solution to the original equation.

Slope

Slope essentially measures the tilt or steepness of a line on a graph. It is reflected as \(m\) in the slope-intercept form \(y = mx + b\). Slope is calculated as the rise (change in \(y\)) over the run (change in \(x\)), often expressed as \(\frac{\text{rise}}{\text{run}}\).

For the equation \(y = 2x - 3\):

• The slope \(m = 2\) means that for every 1 unit you move horizontally, you move 2 units vertically.
• Positive slope values like 2 indicate that the line rises as it moves from left to right.

Knowing the slope helps predict the line's angle without graphing all points, and aids in detecting parallel and perpendicular lines across multiple graphs.

Y-Intercept

In the slope-intercept equation \(y = mx + b\), the y-intercept \(b\) is crucial because it tells you exactly where the line will cross the y-axis. This happens when \(x = 0\). For our specific example:

• In \(y = 2x - 3\), the y-intercept \(b = -3\).
• To plot the y-intercept, place a point at \( (0, -3) \).

Understanding the y-intercept allows you to quickly start plotting the graph. It gives you a fixed point from which you can use the slope to find other points. It serves as the anchor point when sketching a linear graph and is vital for interpreting the equation's relationships in applications.