Problem 10 - Elementary and Intermediate Algebra

Question

Consider the linear function \(y=-\frac{4}{5} x+3\). a. What is the slope of its graph? b. What is the \(y\) -intercept of its graph?

Step-by-Step Solution

Answer

a. The slope is \(-\frac{4}{5}\). b. The \(y\)-intercept is 3.

Step 1: Identifying the Slope

In a linear function given in the form \(y = mx + b\), the coefficient \(m\) represents the slope. For the function \(y = -\frac{4}{5}x + 3\), compare this to the standard form to see that \(m = -\frac{4}{5}\). Thus, the slope is \(-\frac{4}{5}\).

Step 2: Identifying the Y-intercept

In the equation \(y = mx + b\), the constant \(b\) represents the \(y\)-intercept. For the equation \(y = -\frac{4}{5}x + 3\), \(b = 3\). Hence, the \(y\)-intercept is 3.

Key Concepts

Understanding the Slope

The slope of a line is a crucial concept in understanding linear functions. Simply put, the slope tells us how steep the line is. In mathematical terms, the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. It is denoted as \(m\) in the linear equation formula \(y = mx + b\). If the slope \(m = -\frac{4}{5}\), as in our equation, this indicates a few things:

• The line descends from left to right because the slope is negative.
• For every 5 units you move to the right along the x-axis, the line decreases by 4 units on the y-axis.

This is why understanding the slope helps to visualize and interpret the direction and steepness of a line.

The Role of the Y-intercept

The y-intercept is another foundational component of linear equations. It represents the point where the line crosses the y-axis. In the equation \(y = mx + b\), \(b\) denotes the y-intercept. This value indicates the line's starting point, or more technically, the y-value when \(x\) is zero.

• In our given equation, \(y = -\frac{4}{5}x + 3\), the y-intercept \(b\) is 3.
• This means the line crosses the y-axis at the point (0, 3).

Understanding the y-intercept helps in accurately plotting the line on a graph, as it gives a clear starting reference point.

Exploring Linear Equations

Linear equations are equations of the first degree, meaning they involve only linear terms with no exponents or powers other than one. They can be written in various forms, but the most common and easily recognizable is the slope-intercept form: \(y = mx + b\).

• \(m\) represents the slope, which affects the angle and direction of the line.
• \(b\) is the y-intercept, indicating where the line intersects the y-axis.

Linear equations are fundamental in both mathematics and practical applications, as they model relationships where change is constant. They describe situations where two quantities have a linear relationship, which means as one quantity increases or decreases, the other does so at a consistent rate. This makes linear equations a powerful tool in both academic learning and real-world problem solving.