Problem 56 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$4 x+6 y+12=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{2}{3}x - 2\). The slope is -\frac{2}{3} and y-intercept is -2.

Step 1: Rewrite in Slope-Intercept Form

First, isolate \(y\) in the given equation \(4x + 6y + 12 = 0\) to rewrite it in the slope-intercept form. Therefore, subtract \(4x\) and 12 from both sides to get \(6y = -4x - 12\). Finally, divide both sides by 6 to isolate \(y\), obtaining \(y = -\frac{2}{3}x - 2\).

Step 2: Identify the Slope and Y-Intercept

In the equation \(y = -\frac{2}{3}x - 2\), the term multiplying \(x\) is the slope and the constant is the y-intercept. Thus, the slope \(m\) is -\frac{2}{3} and the y-intercept \(b\) is -2.

Step 3: Graph the Equation

Plot the y-intercept point at (-2) on the y-axis. Then, since the slope is negative, go down 2 units and move 3 units to the right from the y-intercept point to mark the second point. Draw a straight line passing through the two points to complete the graph.

Key Concepts

Graphing Linear Equations

The process of graphing linear equations involves creating a visual representation of the equation on a coordinate plane. By plotting points through which the line passes, and connecting these points, the graph of the equation is formed. The most efficient way to graph a linear equation is by using its slope-intercept form, which is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Let's consider how we can apply these steps to the example equation \( 4x + 6y + 12 = 0 \). After rewriting the equation in slope-intercept form as \( y = -\frac{2}{3}x - 2 \), we have a clear view of the slope and the y-intercept. This allows us to begin graphing by first plotting the y-intercept on the y-axis, which is a straightforward step that lays the foundation for the rest of the graphing process.

Once the y-intercept is in place, we use the slope to find additional points. The graphical representation of the equation becomes a straight line through the points that we've plotted. For students, understanding each step of this process ensures that they can graph linear equations encountered in their studies with confidence.

Slope of a Line

The slope of a line measures its steepness and direction. It is typically denoted as \( m \) in the equation of a line in slope-intercept form \( y = mx + b \). Slope is calculated as the ratio of the change in y (rise) over the change in x (run). Therefore, we can interpret slope as how much the line goes up or down for a given horizontal change to the right.

In the context of our example, the slope is \( -\frac{2}{3} \) which can be interpreted as a decline of 2 units in the y direction for every 3 units of increase in the x direction. A negative slope like this indicates that the line tilts downward as one moves from left to right across the graph. Getting a solid grasp of these concepts allows students to visualize and predict the behavior of linear functions, which is a fundamental skill in algebra.

Y-Intercept

The y-intercept is a point where the graph of an equation crosses the y-axis. This happens at \( x = 0 \), which is why it's represented by the constant term \( b \) in the slope-intercept equation \( y = mx + b \). This point provides an important reference for graphing the equation because it is one of the simplest points to locate and plot.

In our example, with the equation rewritten as \( y = -\frac{2}{3}x - 2 \), the y-intercept is -2. Therefore, starting at the origin, you would move down 2 units along the y-axis to plot the y-intercept. From there, understanding the role of the y-intercept in conjunction with the slope gives students a reliable starting point for graphing any linear equation they come across, simplifying what might otherwise seem like a complex procedure.