Problem 55 - College Algebra

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Graph the equation. $$2 x+3 y-18=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the equation is \(y = -2/3x + 6\). The slope of the line is -2/3 and the y-intercept is 6. The graph starts at the y-intercept and moves down 2 units and over 3 units from the y-intercept.

Step 1: Rewrite in Slope-Intercept Form

First, rearrange the equation \(2x + 3y - 18 = 0\) to the form \(y = mx + b\). Begin by moving the term \(2x\) to the right side of the equation: \(3y = -2x + 18\). Then, divide each term by 3 to isolate \(y\): \(y = -2/3x + 6\).

Step 2: Identify the Slope and Y-Intercept

Now that the equation is in slope-intercept form, we can identify the slope and y-intercept. Comparing the equation \(y = -2/3x + 6\) to the general form \(y = mx + b\), we see that \(m = -2/3\) and \(b = 6\). Therefore, the slope of the line is -2/3 and the y-intercept is 6.

Step 3: Graph the Equation

To graph the equation, start by plotting the y-intercept at (0, 6). Since the slope is -2/3, move down 2 units and over 3 units from the y-intercept to find the next point. Draw a line through the points to complete the graph of the equation.

Key Concepts

Linear Equations

Linear equations are foundational elements in mathematics, representing straight lines when graphed on a coordinate plane. These equations are typically written in the form \(Ax + By = C\), where \(x\) and \(y\) are variables, and \(A\), \(B\), and \(C\) are constants. The classic task with linear equations is to express them in the slope-intercept form to facilitate graphing and interpretation.

To rewrite a linear equation into slope-intercept form, \(y = mx + b\), where \(m\) indicates the slope and \(b\) is the y-intercept, you simply solve for \(y\). Adjust the equation by isolating \(y\) on one side of the equation:

• Move terms involving \(x\) and constant terms across the equal sign.
• Divide each term by the coefficient attached to \(y\).

Using these steps, you transform an equation like \(2x + 3y - 18 = 0\) into an easier form to analyze and graph, understanding relationships between the variables.

Graphing Lines

Graphing lines may seem challenging, but it becomes straightforward once you rewrite the equation into the slope-intercept form, \(y = mx + b\). Here, the focus is on finding two fundamental points on the graph:

• The y-intercept, \(b\), where the line crosses the y-axis, is easy to plot since it occurs at the point \((0, b)\).
• A second point that derives from the slope, \(m\), which expresses the rise over the run. For each "rise" (change in \(y\)), you move "run" (change in \(x\)) units along the line.

For instance, with \(y = -\frac{2}{3}x + 6\), the y-intercept is (0, 6). From this point, apply the slope \(-\frac{2}{3}\) by moving down 2 units vertically and 3 units horizontally. This gives a second point, which together with the intercept, helps you draw the straight line representing the equation.

Slope and Y-Intercept

The slope and y-intercept are two key elements of a line in a coordinate plane. The slope, denoted as \(m\), indicates how steep the line is. It's calculated as the ratio of the vertical change to the horizontal change between two points on the line, often referred to as "rise over run."

The y-intercept, represented by \(b\), is where the line crosses the y-axis. It is easy to identify once the equation is in slope-intercept form, \(y = mx + b\), appearing simply as the constant term.

In our example equation, \(y = -\frac{2}{3}x + 6\):

• Slope \(m = -\frac{2}{3}\): This negative value shows the line tilts downwards from left to right.
• Y-intercept \(b = 6\): The line crosses the y-axis at this point, making it a critical starting point for graphing the line.

Understanding these terms and how they relate facilitates graphing and interpreting the behavior of linear relationships.