Problem 39 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. With slope \(-\frac{1}{2},\) through \(\left(0, \frac{5}{3}\right)\)

Step-by-Step Solution

Answer

The equation is \( y = -\frac{1}{2}x + \frac{5}{3} \).

Step 1: Understand 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. Here, the slope \( m \) is given as \( -\frac{1}{2} \).

Step 2: Identify the Y-Intercept

Since the line passes through the point \( \left(0, \frac{5}{3}\right) \), this point is the y-intercept \( b \). Thus, \( b = \frac{5}{3} \).

Step 3: Write the Equation

Use the slope \( m = -\frac{1}{2} \) and y-intercept \( b = \frac{5}{3} \) in the slope-intercept form. The equation of the line becomes \( y = -\frac{1}{2}x + \frac{5}{3} \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed on a coordinate plane. These equations can be expressed in various forms, with the slope-intercept form being one of the most common. In general, a linear equation in this form appears as \( y = mx + b \), where \( y \) is the dependent variable that represents the value on the vertical axis. The \( x \) is the independent variable that represents the value on the horizontal axis. Linear equations describe relationships where the change between two variables happens at a constant rate. This predictable pattern is crucial for finding solutions to many real-world problems, such as determining profit over time, or predicting scores in athletics.
Here's a quick reminder:

• The graph of a linear equation forms a straight line.
• Every point on this line is a solution to the equation.
• A linear equation can represent various real-world situations and problems.

Slope of a Line

The slope of a line is a measure that tells us how steep the line is, essentially describing the line's angle with respect to the horizontal axis. Mathematically, the slope is represented as the ratio \( \frac{\text{change in } y}{\text{change in } x} \) or "rise over run." It indicates how much the \( y \)-value changes for each unit that the \( x \)-value changes. This property allows us to determine not only how a line appears but also how values associated with real-life applications vary.

• Positive slope: As \( x \) increases, \( y \) also increases, meaning the line rises to the right.
• Negative slope: As \( x \) increases, \( y \) decreases, so the line falls to the right.
• A slope of zero indicates a horizontal line.

You might encounter slopes as fractions, whole numbers, or even decimals, but they all depict the same idea of how two quantitative variables are related. For instance, a slope of \(-\frac{1}{2}\) means for every increase by 1 in \( x \), the \( y \)-value decreases by 0.5.

Y-Intercept

The y-intercept is a key component of the slope-intercept form of a linear equation. It describes the point where the line crosses the \( y \)-axis. In practical terms, if you imagine the graph of a line, you can find the y-intercept by looking at where this line meets the vertical axis. In the equation \( y = mx + b \), the \( b \) value is the y-intercept.

This constant shows us the starting value for \( y \) when \( x \) is zero, offering insights into the initial condition or starting point of the linear relationship.

• The line crosses the \( y \)-axis at the y-intercept point \( (0, b) \).
• This tells us the "baseline" \( y \)-value when there is no contribution from the \( x \)-changes.
• It provides a crucial anchor point for graphing the line.

In our problem, since the line passes through the point \((0, \frac{5}{3})\), \( b = \frac{5}{3} \). This means when \( x = 0 \), \( y \) is \( \frac{5}{3} \), providing the starting value of the line.