Problem 21 - Algebra and Trigonometry

Question

Find an equation of the line that satisfies the given conditions. Slope \(3 ; \quad y\) -intercept \(-2\)

Step-by-Step Solution

Answer

Equation: \( y = 3x - 2 \).

Step 1: Understanding the Slope-Intercept Form

The slope-intercept form of the equation of a line is given by \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

Step 2: Identifying Parameters

From the given conditions, the slope \( m \) is 3, and the y-intercept \( b \) is -2.

Step 3: Substitute the Parameters into the Equation

Using the identified slope and y-intercept, plug these values into the slope-intercept equation: \[ y = 3x - 2 \].

Key Concepts

slope-intercept form

The slope-intercept form is an essential concept in algebra that provides a straightforward way to write the equation of a straight line. This form is represented as \( y = mx + b \). Here, \( y \) is the dependent variable, changing according to \( x \), the independent variable. Understanding this equation is vital because it clearly shows two critical elements of a line: its slope and its y-intercept.

By using the slope-intercept form, you can quickly determine how a line behaves on a graph. To utilize this form effectively, it is crucial to recognize the meanings of \( m \) and \( b \) and how they influence the graph's appearance. The slope \( m \) indicates the steepness and direction of the line, while \( b \) provides the point where the line crosses the y-axis.

This form is commonly used due to its simplicity and ease of application, making it an ideal choice for solving problems involving linear equations.

slope

The slope of a line is a number that describes how steep or flat the line is. It is denoted by \( m \) in the slope-intercept form \( y = mx + b \). The slope tells you how much \( y \) changes for a unit change in \( x \). For example, a slope of 3 means that for each step you move to the right by 1 on the x-axis, the line rises by 3 on the y-axis.

Understanding slope involves recognizing the two possible directions it can indicate:

• If \( m \) is positive, the line slopes upwards from left to right.
• If \( m \) is negative, the line slopes downwards from left to right.

The greater the value of \( m \), the steeper the line. Conversely, smaller values suggest a gentler slope. To understand and calculate the slope, you should think of it as "rise over run," which means how much upward (or downward) change there is for a horizontal movement.

Slopes are fundamental in helping you predict and graph linear equations, allowing for the analysis of how particular changes in \( x \) influence \( y \).

y-intercept

The y-intercept is a vital concept in understanding linear equations and graphing lines. It is represented by \( b \) in the slope-intercept form \( y = mx + b \). The y-intercept refers to the point where the line crosses the y-axis. This occurs when \( x \) equals zero. For the given equation \( y = 3x - 2 \), the y-intercept is \(-2\), meaning the line will cross the y-axis at the point (0, -2).

Interpreting the y-intercept can help in visualizing the position of the line on a graph. Knowing this point simplifies graphing a line because it gives you a specific point to start from. It also gives insight into the line's behavior when the value of \( x \) is zero.

Overall, the y-intercept is crucial for understanding the starting point of a linear equation within the broader context of a graph. This value, combined with the slope, provides a complete picture of the line's path.