Problem 1 - 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. $$y=3 x+2$$

Step-by-Step Solution

Answer

The slope of the line is 3 and the y-intercept is 2.

Step 1: Identify the slope

The equation of the line is in the form \(y = mx + c\). The coefficient of \(x\) is the slope of the line. Thus, in the equation \(y = 3x + 2\), the slope \(m\) is 3.

Step 2: Identify the y-intercept

In the equation of the line \(y = mx + c\), \(c\) is the y-intercept. Thus, in the equation \(y = 3x + 2\), the y-intercept is 2.

Key Concepts

Understanding Slope

The slope of a line in a linear equation indicates the line's steepness and the direction it moves on a graph. It is often represented by the letter "m". In a mathematical context, we define slope as the ratio of the vertical change to the horizontal change between two distinct points on a line.

Practically speaking, the slope tells us how much "rise" (change in y) there is for a given "run" (change in x). For example, a slope of 3, as seen in the equation \(y = 3x + 2\), means that for every unit you move to the right on the x-axis, the y-value increases by 3 units. This results in a line tilting upward from left to right, signifying a positive slope.

A few key concepts regarding slope include:

• A positive slope moves upwards from left to right.
• A negative slope descends from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope corresponds to a vertical line.

Understanding the slope allows for a deeper comprehension of how different linear equations represent unique lines on a graph.

Deciphering the Y-Intercept

The y-intercept reflects the point where a line crosses the y-axis on a graph. In the equation \(y = mx + c\), the y-intercept is denoted by the constant \(c\). This concept is crucial as it sets the starting point of the line when x equals zero.

In the example given \(y = 3x + 2\), the y-intercept is 2. This means that the line intersects the y-axis at the point (0, 2). In a graph, you will notice the line starts at this point and extends in the direction determined by its slope.

Knowing the y-intercept provides information on where the line starts in relation to the y-axis, and matters significantly when graphing:

• The y-intercept is often the initial value when x is zero.
• It offers a quick reference for graphing, showing where the line begins on the y-axis.
• Changes in the y-intercept value shift the line up or down the graph.

Linear Equations Unraveled

A linear equation is an equation that models a straight line when plotted on a graph. The standard form of these equations is often expressed as \(y = mx + c\), where \(m\) represents the slope and \(c\) represents the y-intercept.

Let's make sense of this with an example: the equation \(y = 3x + 2\). Here, 3 is the slope, and 2 is the y-intercept. These values direct you to draw a line that slopes upwards (thanks to the positive slope of 3) and begins on the y-axis at the point (0, 2).

Describing linear equations involves understanding:

• The coefficient of \(x\) (the slope) shows how steep the line is and in which direction it tilts.
• The constant \(c\) (the y-intercept) shows where the line crosses the y-axis.
• These equations always form straight lines.
• They are foundational for more complex mathematics in algebra and calculus.

Mastering linear equations and their components - slope and y-intercept - is a crucial step in understanding more advanced mathematical concepts.