Problem 3 - 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. $$y=3 x-5$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Slope

Looking at the equation \(y = 3x - 5\), the coefficient of \(x\) is the slope of the line. In this case, the slope \(m\) is 3.

Step 2: Identify the Y-Intercept

The y-intercept \(b\) is the constant value in the equation, which is the point where the line crosses the y-axis. Here, the y-intercept is -5.

Key Concepts

Slope

The slope of a line is a crucial part of understanding linear equations. It measures how steep a line is. Imagine you are hiking on a trail; the slope would represent how steep the hill is. In mathematical terms, the slope refers to the "rise over run" of a line. That means it compares the vertical change (up and down) to the horizontal change (side to side).

The slope, denoted as \(m\), essentially tells us how much \(y\) increases or decreases as \(x\) increases by one unit. In the equation of a line like \(y = mx + b\), the slope is the coefficient of \(x\). For our line \(y = 3x - 5\), the slope \(m\) is 3.

Here's a quick summary:

• Slope is the ratio of change in \(y\) to change in \(x\).
• It can be positive, negative, zero, or undefined.
• In \(y = 3x - 5\), the slope of 3 indicates a steepness of line and that the line rises 3 units up for every 1 unit it moves to the right.

This understanding of slope is important for predicting how changes in \(x\) affect \(y\). You can visualize it as how fast or slow the graph of the line ascends or descends.

Y-Intercept

The y-intercept is where a line touches or crosses the y-axis on a graph. It's the value of \(y\) when \(x\) is zero. In essence, it's where the line starts when \(x\) is zero. In the linear equation \(y = mx + b\), the y-intercept is the constant term \(b\).

For the equation \(y = 3x - 5\), the y-intercept is -5. This means when \(x\) equals 0, \(y\) will be -5, pointing to the line crossing the y-axis at -5 on the graph.

A few important points about y-intercepts:

• The y-intercept is always on the y-axis of the graph.
• It provides a starting point for drawing the line.
• Depending on its value, it tells us how high or low the line is vertically shifted on the graph.

Understanding the concept of y-intercept helps in plotting the line quickly on a graph, showing where the line begins in relation to the y-axis.

Equation of a Line

An equation of a line gives you a complete picture of how the line behaves on a cartesian coordinate system. In the form \(y = mx + b\), it is often called the slope-intercept form, as it provides both the slope \(m\) and the y-intercept \(b\). This makes it very intuitive for graphing and understanding linear relationships.

For instance, in our example equation \(y = 3x - 5\), we have all the information we need:

• The slope \(m = 3\) tells us the direction and steepness of the line.
• The y-intercept \(-5\) informs us where the line crosses the y-axis.

Such an equation makes it easy to calculate \(y\) for any given \(x\). It guides you in plotting the line step-by-step:

1. Start at the y-intercept (0, -5).
2. Use the slope to determine the next points. For a slope of 3, go up 3 units and right 1 unit repeatedly.

Understanding this form is key for anyone working with graphs and helps build a strong foundation in algebraic concepts and linear relationships.