Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) $$\text { Through }(-8,1), m=-0.5$$

Step-by-Step Solution

Answer

The equation is \( y = -0.5x - 3 \).

Step 1: Identifying 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. We need to find the value of \( b \) to complete the equation for the given line.

Step 2: Substitute Given Values

We know the slope \( m = -0.5 \) and the point \( (-8, 1) \) that the line passes through. Substitute these values into the slope-intercept equation: \( 1 = -0.5(-8) + b \).

Step 3: Solve for the Y-Intercept

Perform the multiplication: \( -0.5 \times -8 = 4 \). Substitute back into the equation: \( 1 = 4 + b \). To solve for \( b \), subtract 4 from both sides: \( b = 1 - 4 \).

Step 4: Calculate the Y-Intercept

Simplify the equation from the previous step to find \( b = -3 \). Now we have all the information to write the full equation of the line.

Step 5: Write the Final Equation

Insert the values of \( m \) and \( b \) into the slope-intercept form: \( y = -0.5x - 3 \). This is the equation for the line that passes through \( (-8, 1) \) with a slope of \( -0.5 \).

Key Concepts

Linear Equations

Linear equations are fundamental in understanding how straight lines work on a coordinate plane. When we talk about linear equations, we usually recognize them by their standard form, \[ ax + by = c \]or, more commonly, the slope-intercept form: \[ y = mx + b. \]The beauty of linear equations lies in their simplicity and how they're used to describe relationships where there is a constant rate of change, known as the "slope." This form is very user-friendly and helps to easily determine both the slope and the y-intercept of the line.In any linear equation:

• \( x \) and \( y \) represent variables whose values are determined depending on the values of each other.
• The coefficients \( m \) and the constant \( b \) hold crucial information, standing for the slope and y-intercept, respectively.

Understanding linear equations is crucial because they appear in various real-world situations, such as calculating distances, predicting expenses, and in data analysis models.
Their predictability makes them highly reliable for establishing relationships between variables.

Slope

The slope is a critical component of linear equations as it defines the direction and steepness of a line. The slope is represented by \( m \) in the slope-intercept formula \( y = mx + b \). You can think of the slope as "rise over run," showing the change in the vertical direction for each unit of horizontal change. For instance, if \( m = -0.5 \), the line moves down half a unit vertically for every one unit it moves horizontally.The slope can tell you a lot about the line:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero means the line is horizontal, showing no vertical change.
• An undefined slope (division by zero) means the line is vertical.

The slope gives us a clear understanding of how two quantities relate to each other. In terms of real-world applications, knowing the slope of a line can immediately tell you how one variable affects another, which is crucial for analysis in various fields like physics, economics, and engineering.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form equation, \( y = mx + b \), the y-intercept is represented by \( b \).This is the value of \( y \) when \( x = 0 \). Think of it as the starting point of the line on the graph. For the line described in the exercise, where the equation we determined is \( y = -0.5x - 3 \), the y-intercept is \(-3\). Understanding the y-intercept is helpful because:

• It gives you one easy-to-find point on the line.
• It serves as a reference point for drawing the rest of the line by using the slope.
• In real-life situations, it often represents the initial value before any changes occur. For example, in a budget, it might represent fixed costs.

When plotting graphs, knowing the y-intercept allows you to know where to start, making it easier to understand the behavior of the line right from the beginning. The y-intercept is a vital element that helps make sense of how the linear equation represents different situations or data.