Problem 57 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$8 x-4 y-12=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \(y = 2x - 3\). The slope is 2 and the y-intercept is -3.

Step 1: Rewriting into Slope-Intercept Form

Start by rearranging the given equation \(8 x-4 y-12=0\) to the form \(y=mx+b\). This is done by isolating \(y\). Firstly, rearrange terms to get \(-4y = -8x + 12\). Then, divide every term by -4 to solve for \(y\). This results in \(y = 2x - 3\).

Step 2: Determining the Slope and Y-Intercept

In the equation \(y = 2x - 3\), m which is the coefficient of \(x\) gives the slope of the line, and \(b\) is the y-intercept. Thus, the slope (m) is 2 and the y-intercept (b) is -3.

Step 3: Graphing the Equation

To graph \(y = 2x - 3\), start at the y-intercept (0, -3) on the y-axis. From there, use the slope to find the next point. In this case, with slope of 2, move 2 units up and 1 unit to the right to arrive at the next point (1, -1). Continue this pattern to draw the line.

Key Concepts

Slope-Intercept Form

When working with linear equations, one of the most common forms you'll encounter is the slope-intercept form. This form is written as \(y = mx + b\). In this equation, "\(m\)" represents the slope of the line, while "\(b\)" denotes the y-intercept. The slope-intercept form is highly useful because it quickly tells you both the steepness of the line and where it crosses the y-axis. To convert a given linear equation into the slope-intercept form, you'll want to rearrange it so \(y\) is isolated on one side of the equation. Let's say you start with an equation like \(8x - 4y - 12 = 0\). Rearrange it by moving terms to isolate \(y\) to find \(y = 2x - 3\). Now, you're all set. The equation is now in slope-intercept form.

Slope of a Line

The slope of a line is a measure of its steepness or incline. It is symbolized by 'm' in the equation \(y = mx + b\). Usually, the slope is calculated as the "rise" over the "run." This means you look at how much the line goes up or down for each step to the right along the graph. In our example equation \(y = 2x - 3\), the slope \(m = 2\). So, every time you move one unit to the right, you'll move two units up. Understanding the slope is crucial because it tells us how the line behaves. A positive slope means the line ascends from left to right, while a negative slope indicates it descends as you move right. With a slope of 2, this line is moderately steep.

Y-Intercept

The y-intercept of a line is the point where it crosses the y-axis on a graph. It is denoted by 'b' in the equation \(y = mx + b\). The y-intercept is important because it gives you a specific starting point when graphing your line. In our equation \(y = 2x - 3\), the y-intercept is \(-3\). This means that if you set \(x = 0\), you will find the point \((0, -3)\). This point is where your line will intersect the y-axis. Knowing this point is essential for accurately graphing any linear equation since it provides a clear reference for the placement of your line.

Linear Equation Graphing

Graphing a linear equation involves plotting points on a coordinate plane and drawing a line through these points. When you have an equation in slope-intercept form, like \(y = 2x - 3\), it's straightforward to graph. Start by locating the y-intercept, which in this case is -3, so place a point at (0, -3) on the y-axis. Next, use the slope to find another point. For a slope of 2, move two units up and one unit to the right from the y-intercept to reach (1, -1). Continue this pattern to establish a line. Graphing becomes intuitive once you understand slope and y-intercept. This visual representation allows you to see the behavior and direction of the line clearly.