Problem 28 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. \(x\) -intercept \(-6, y\) -intercept \(-8\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y = -\frac{4}{3}x - 8\).

Step 1: Understand the Problem

You are tasked with finding the slope-intercept form of a line given the x-intercept and y-intercept. The x-intercept of -6 implies the point \((-6, 0)\), and the y-intercept of -8 implies the point \((0, -8)\).

Step 2: Recall the Slope-Intercept Form

The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Calculate the Slope

The slope \(m\) of a line that passes through points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points \((-6, 0)\) and \((0, -8)\), apply the formula: \[ m = \frac{-8 - 0}{0 - (-6)} = \frac{-8}{6} = -\frac{4}{3} \].

Step 4: Write the Equation Using Slope and Y-Intercept

Using the slope \(-\frac{4}{3}\) and the y-intercept \(-8\), substitute these values into the slope-intercept form: \(y = -\frac{4}{3}x - 8\).

Step 5: Final Check

Verify that the equation \(y = -\frac{4}{3}x - 8\) is satisfied by both the x-intercept and y-intercept. When \(x = -6\), \(y = 0\), and When \(y = -8\), \(x = 0\). Hence, the equation is correct.

Key Concepts

Understanding the x-intercept

The x-intercept of a line is the point where the line crosses the x-axis. At this point, the value of the y-coordinate is zero. In our problem, the x-intercept is given as -6, which is represented by the point \((-6, 0)\). This means that when \(x = -6\), the value of \(y\) is zero.
To determine the x-intercept:

• Set \(y = 0\) in the line equation.
• Solve for \(x\) to find the point.

This intercept provides a crucial point that helps in constructing the equation of the line when paired with the y-intercept.

Exploring the y-intercept

The y-intercept is the point where a line crosses the y-axis. At this point, the x-coordinate is always zero. In our case, the y-intercept is -8, which is expressed as the point \((0, -8)\). So, when \(x = 0\), \(y\) equals -8.
To identify the y-intercept:

• Set \(x = 0\) in the line equation.
• Solve for \(y\), which will give the y-intercept.

Utilizing the y-intercept in the slope-intercept form equation \(y = mx + b\) is straightforward, as \(b\) corresponds to the y-intercept value.

Mastering slope calculation

The slope of a line is a measure of its steepness or incline and is crucial in forming the equation of the line in slope-intercept form. It is derived from two points on the line using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
To compute the slope:

• Identify the points, in this exercise, they are \((-6, 0)\) and \((0, -8)\).
• Substitute into the slope formula: \[ m = \frac{-8 - 0}{0 - (-6)} = \frac{-8}{6} = -\frac{4}{3} \].

The resulting slope, \(-\frac{4}{3}\), shows a line leaning downward, moving rightward across the graph. This slope value is vital as it determines the angle and direction of your line.