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Chapter 1: Problem 5

Fill in the blank. The equation \(y=m x+b\) is the ________ form of the equation of a line.

Short Answer

Expert verified
slope-intercept

Step by step solution

01

Identify the equation format

The given equation is of the form: y = mx + b
02

Recognize the components

In this equation, 'm' represents the slope of the line, and 'b' represents the y-intercept, which is where the line crosses the y-axis.
03

Determine the type of equation

The form of the equation y = mx + b is known as the slope-intercept form because it directly shows the slope and the y-intercept of the line.
04

Fill in the blank

Thus, the equation y = mx + b is the slope-intercept form of the equation of a line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Line
An equation of a line is a mathematical expression that describes all the points along the line in a coordinate plane. One of the most common forms of this equation is the slope-intercept form. This form allows us to easily understand and graph the line using just two components: the slope and the y-intercept. By recognizing these parts, we can quickly identify how the line behaves and where it crosses the y-axis. So, understanding the equation of a line is crucial for studying linear relationships in mathematics.
Slope
The slope of a line, denoted by 'm' in the slope-intercept form equation, measures the steepness and direction of the line. It is calculated as the ratio of the change in y (vertical change) to the change in x (horizontal change) between two points on the line. The formula for slope is \(\frac{\text{rise}}{\text{run}}\) or \(\frac{y_2 - y_1}{x_2 - x_1}\). A positive slope means the line is increasing, while a negative slope means the line is decreasing. A zero slope represents a horizontal line, and an undefined slope (division by zero) represents a vertical line.
y-Intercept
The y-intercept of a line is the value at which the line crosses the y-axis. In the slope-intercept form equation \(y = mx + b\), the y-intercept is represented by 'b'. This means when x is 0, y equals b. It's useful because it gives a starting point for graphing the line. You can always start at the y-intercept and use the slope to find other points on the line.
Linear Equation
A linear equation is an equation that graphs a straight line. The general form of a linear equation in two variables is \(Ax + By = C\), but the slope-intercept form \(y = mx + b\) is often more convenient. Linear equations are fundamental in algebra as they model relationships with a constant rate of change. These equations and their graphs are foundational in understanding more complex mathematical concepts, so mastering them will improve comprehension of the broader subject matter.

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