Problem 3 - Introductory Algebra for College Students

Question

Find the slope and the \(y\) -intercept of the line with the given equation. $$y=3 x-5$$

Step-by-Step Solution

Answer

The slope of the line is 3 and the y-intercept is -5.

Step 1: Identify the Slope \(m\)

The slope of the line is the coefficient of \(x\) in the equation, which in this case is 3. So, the slope of the line is 3.

Step 2: Identify the y-intercept \(b\)

The y-intercept of the line is the constant term in the equation, which in this case is -5. So, the y-intercept of the line is -5.

Step 3: Express the Results

The slope of the line is 3 and the y-intercept is -5.

Key Concepts

Understanding the Slope

The slope of a line in the context of linear equations is a measure of the line's steepness or incline. It tells us how much the line rises or falls as we move along the x-axis. In the equation given, the slope is represented by the coefficient of \(x\). For the equation \(y = 3x - 5\), the slope is 3. This means for every one unit increase in \(x\), the value of \(y\) increases by 3 units. Understanding slope is crucial because it gives us insight into the rate of change between two variables. The slope can be calculated using the formula \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in \( y \) and \( \Delta x \) is the change in \( x \). In everyday language, slope is like the speed of the line: the larger the absolute value of the slope, the steeper the line.

• If the slope is positive, the line rises.
• If the slope is negative, the line falls.
• If the slope is zero, the line is horizontal.

These interpretations help in visualizing how the line behaves on the graph.

Exploring the Y-Intercept

The \( y \)-intercept is a fundamental aspect of a linear equation. It is where the line crosses the y-axis on a graph. When looking at any linear equation in the form \(y = mx + b\), the \( y \)-intercept is indicated by the constant term \(b\). In our example \(y = 3x - 5\), the \( y \)-intercept is -5. This means when \(x = 0\), the value of \(y\) is -5. The \( y \)-intercept serves as the starting point of the line when graphing. No matter the slope of the line, the \( y \)-intercept remains fixed on the \( y \)-axis.

• It can be thought of as the value of \(y\) when no effect of \(x\) is present.
• It is essential in describing how the line sits in relation to the origin of the graph.

By focusing on the \( y \)-intercept, you can easily place the line on the graph and then use the slope to plot the direction and steepness.

Decoding Linear Functions

Linear functions, like \(y = 3x - 5\), describe straight lines through a coordinate plane and are foundational in understanding algebraic expressions. They can often model real-world situations where there is a constant rate of change. Such functions are characterized by their constant slope and simple structure. A linear function can be expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. These two elements govern the entire behavior of the line:

• The slope \(m\) indicates the direction and steepness of the line.
• The \(y\)-intercept \(b\) shows the initial value when \(x\) is zero.

Linear functions are one of the simplest forms of functions but pivotal for building a foundation in mathematics. Understanding them allows students to approach more complex topics with ease, as well as apply these basics to graphical and practical scenarios.