Problem 4 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-12,\) find the slope and the \(y\) -intercept of the line with the given equation. $$y=4 x-2$$

Step-by-Step Solution

Answer

The slope of the line is \(4\), and the y-intercept is \(-2\).

Step 1: Identify the Slope

The number in front of \(x\) in the equation \(y=4x-2\) is the slope. This number is \(4\). Therefore, the slope of the line is \(4\).

Step 2: Identify the y-intercept

The number at the end of the equation \(y=4x-2\) is the y-intercept. This number is \(-2\). Therefore, the y-intercept of the line is \(-2\).

Key Concepts

Linear Equations

Linear equations are a fundamental concept in algebra, representing lines in a two-dimensional space. Every linear equation can be written in the slope-intercept form: \( y = mx + b \). In this format:

• \(y\) represents the dependent variable, often corresponding to the vertical axis in a graph.
• \(x\) is the independent variable, usually associated with the horizontal axis.
• \(m\) stands for the slope of the line.
• \(b\) represents the y-intercept.

Understanding linear equations allows us to easily graph them, solve problems, and interpret data trends. When working with these equations, identifying the slope and y-intercept helps to understand the line's direction and initial point.

Slope

The slope of a line indicates how steep the line is and the direction it goes. It tells us how much \(y\) changes for a unit change in \(x\). Specifically, it is the "rise over run," or the change in \(y\) divided by the change in \(x\). In the equation given, \(y = 4x - 2\), the slope is \(4\).

• 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.
• If the slope is zero, the line is horizontal.
• An undefined slope applies to vertical lines, which do not fit the slope-intercept form.

Knowing the slope helps determine the angle of the line as it intersects with the axes, aiding in making predictions from the graph.

y-intercept

The y-intercept is where the line crosses the y-axis. This intercept helps us determine the point where \(x = 0\). In our example equation, \(y = 4x - 2\), the y-intercept is \(-2\). This means the line passes through the point \( (0, -2) \).

• The value of \(b\) in the equation represents this point on the graph.
• When graphing, you start plotting the line at this intercept point.
• It serves as the anchor point for drawing the line based on the slope.

Recognizing the y-intercept is crucial for accurate graph formation and helps in understanding where the line begins on the graph.