Problem 31 - College Algebra with Modeling and Visualization

Question

Find the slope-intercept form for the line satisfying the conditions. Passing through \((0,-6)\) and \((4,0)\)

Step-by-Step Solution

Answer

The equation is \( y = \frac{3}{2}x - 6 \).

Step 1: Understand the Slope-Intercept Form

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

Step 2: Calculate the Slope (m)

The slope \(m\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the given points \((0, -6)\) and \((4, 0)\) into the formula: \[ m = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2} \].Thus, the slope is \(\frac{3}{2}\).

Step 3: Identify the Y-intercept (b)

Since the point \((0, -6)\) lies on the line, the y-coordinate of this point is the y-intercept \(b\). Hence, \(b = -6\).

Step 4: Write the Slope-Intercept Form Equation

Now that we have both the slope \(m\) and the y-intercept \(b\), we can write the equation of the line as:\[ y = \frac{3}{2}x - 6 \].

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra and represent lines on a graph. They are expressed in various forms, with the slope-intercept form being one of the most commonly used. In the slope-intercept form, the equation of the line is written as \(y = mx + b\). Here, \(x\) and \(y\) are variables that represent any point on the line, and \(m\) and \(b\) are constants.

• \(m\) is the slope of the line, determining how steep the line is.
• \(b\) is the y-intercept, which is where the line crosses the y-axis.

Linear equations are crucial because they allow us to describe relationships between two variables and make predictions about one variable based on another. They are the type of equations that form straight lines when graphed. Understanding how to manipulate these equations helps in solving real-world problems and interpreting data.

Slope

The slope of a line is a measure of its steepness. It tells us how much the line rises (or falls) for each unit that it runs horizontally. Mathematically, slope is defined as the ratio of the change in the y-values to the change in the x-values between two points on the line. The formula for slope \(m\) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In our problem, the points are \((0, -6)\) and \((4, 0)\). By substituting these coordinates into the slope formula, we find:
\[m = \frac{0 - (-6)}{4 - 0} = \frac{6}{4} = \frac{3}{2}\]Thus, the slope \(m\) of the line passing through these points is \(\frac{3}{2}\).
A positive slope indicates the line rises as you move from left to right, while a negative slope means the line falls.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation \(y = mx + b\), the y-intercept is represented by \(b\). Understanding the y-intercept is key because it represents the value of \(y\) when \(x\) is zero.
In our exercise, one of the given points is \((0, -6)\). This tells us directly that the y-intercept \(b\) is \(-6\). When an equation is in slope-intercept form, it can be quickly read to determine both the slope and the y-intercept.

• The y-intercept answers the question, "Where does the line meet the y-axis?"
• It's useful for graphing because it provides a starting point for drawing the line.

By clearly identifying \(b\), you can describe the line fully along with its slope, making it easier to understand and work with.