Problem 35 - College Algebra

Question

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. \(x\) -intercept at (-2,0) and \(y\) -intercept at (0,-3)

Step-by-Step Solution

Answer

The linear equation is \( y = \frac{-3}{2}x - 3 \).

Step 1: Identify the given intercepts

We are given the points where the line crosses the x-axis and y-axis. The x-intercept is (-2, 0) and the y-intercept is (0, -3).

Step 2: Determine the slope

The slope (m) of the line is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Substituting in the intercepts: \[ m = \frac{-3 - 0}{0 - (-2)} = \frac{-3}{2} \].

Step 3: Write the equation in slope-intercept form

Using the slope-intercept form of the line equation, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We know \( m = \frac{-3}{2} \) and the y-intercept is -3, thus: \[ y = \frac{-3}{2}x - 3 \].

Step 4: Verify the equation with the x-intercept

Substitute x = -2 into the equation \( y = \frac{-3}{2}x - 3 \) to verify the x-intercept: \[ y = \frac{-3}{2}(-2) - 3 = 3 - 3 = 0 \].The equation satisfies the x-intercept at (-2, 0).

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is the point where the line crosses the x-axis. In other words, it is the x-coordinate of the point when the y-value is zero. For example, in the given exercise, the x-intercept is (-2, 0), meaning the line passes through the point (-2, 0) on the x-axis. It is crucial because it provides a specific point that can help plot the line.
To identify the x-intercept from a given linear equation like \( y = mx + b \), set \( y = 0 \) and solve for \( x \). This intercept is useful for graphing because it's one of the fixed points that define the line's behavior and slope on a graph.

Finding the Y-Intercept

A y-intercept is where the line crosses the y-axis. This occurs when the x-coordinate is zero, leaving only the y-value to identify where the crossing happens. In our problem, the y-intercept is (0, -3). The y-intercept can often be found directly as the constant \( b \) in the slope-intercept form equation \( y = mx + b \). Here, \( b = -3 \), indicating that the line will meet the y-axis at the point (0, -3).
Graphically, the y-intercept is an easily accessible starting point for drawing the line, as it helps illustrate where the line begins its path through the plane.

Using Slope-Intercept Form

The slope-intercept form of a linear equation is \( y = mx + b \). This format is beneficial because it clearly defines the slope \( m \) and the y-intercept \( b \). For the equation derived from the exercise, we have \( y = \frac{-3}{2}x - 3 \). Here, the slope \( m \) is \( \frac{-3}{2} \), indicating the steepness and direction of the line, while the y-intercept \( b \) is -3.
The slope-intercept form makes it straightforward to graph linear equations, as you can start at the y-intercept and use the slope to find subsequent points on the line by rising or falling and moving left or right across the grid based on the slope's fraction.

Calculating the Slope

Calculating the slope \( m \) is fundamental to understanding a linear equation. The formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) calculates how much the line rises vertically relative to how much it runs horizontally. Given the intercepts (-2, 0) and (0, -3), plug them into the slope formula:

• \( y_1 = 0 \)
• \( y_2 = -3 \)
• \( x_1 = -2 \)
• \( x_2 = 0 \)

Thus, \( m = \frac{-3 - 0}{0 - (-2)} = \frac{-3}{2} \). The slope of \(-\frac{3}{2}\) implies that for every 2 units you move horizontally on the graph, the line descends by 3 units vertically. This characteristic helps indicate the direction in which the line tilts and how steep it is.