Problem 41 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in standard form unless indicated otherwise. See Examples 1 through \(6 .\) With slope \(-\frac{1}{2} ; y\) -intercept \(\left(0, \frac{3}{8}\right) ;\) use slope-intercept form.

Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{2}x + \frac{3}{8} \).

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.

Step 2: Identify Given Values

From the problem statement, the slope \( m \) is \(-\frac{1}{2}\) and the y-intercept \( b \) is \(\frac{3}{8}\). These values will be used directly in the slope-intercept form.

Step 3: Substitute Values into Slope-Intercept Form

Replace \( m \) and \( b \) in the equation \( y = mx + b \) with \(-\frac{1}{2}\) and \(\frac{3}{8}\) respectively. This gives us the equation: \( y = -\frac{1}{2}x + \frac{3}{8} \).

Step 4: Finalize the Equation

The equation \( y = -\frac{1}{2}x + \frac{3}{8} \) is already in slope-intercept form, which matches the request from the problem statement.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express linear relationships. This format is given by the equation \( y = mx + b \). Here is what each symbol represents:

• \( y \): The value of the dependent variable (often on the vertical axis).
• \( m \): The slope of the line, indicating its steepness and direction.
• \( x \): The value of the independent variable (often on the horizontal axis).
• \( b \): The y-intercept, or where the line crosses the y-axis.

Using slope-intercept form makes it easy to quickly identify the slope and y-intercept of a line. This is especially handy when you need to graph the equation or understand the relationship between variables.

Slope

Slope is a measure of the steepness or incline of a line. In mathematical terms, the slope \( m \) is a ratio that describes how much \( y \) increases or decreases as \( x \) changes. It's calculated as the change in \( y \) divided by the change in \( x \), often referred to as "rise over run."

For example:

• If the slope \( m \) is \(-\frac{1}{2}\), this means that for every increase of 2 units in \( x \), \( y \) decreases by 1 unit.
• A positive slope means the line rises from left to right, while a negative slope means it falls.

Understanding the slope allows us to predict how changes in one variable affect another, making it a useful concept in various applications like economics and physics.

Y-Intercept

The y-intercept \( b \) is a crucial part of the equation, as it determines where the line crosses the y-axis. In the equation \( y = mx + b \), it is represented by \( b \).

Here's a closer look at the y-intercept:

• When \( x \) is zero, the y-value is \( b \). This is the point on the graph where the line hits the y-axis.
• For example, if the y-intercept is \( \frac{3}{8} \), then when \( x \) is zero, \( y \) will be \( \frac{3}{8} \).
• Knowing the y-intercept helps us start drawing the line accurately from the y-axis.

The y-intercept provides a starting point for graphing a line and offers insight into the behavior of the equation at \( x = 0 \).