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Chapter 12: Problem 66

What is the slope-intercept form of the equation of a line?

Short Answer

Expert verified
The slope-intercept form is \( y = mx + b \) where \( m \) is the slope, and \( b \) is the y-intercept.

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is where the line crosses the y-axis.
02

Identify Components of the Equation

In the slope-intercept form \( y = mx + b \), the variable \( m \) corresponds to the slope. The slope describes the steepness and direction of the line. The \( b \) represents the y-intercept, indicating the point at which the line crosses the y-axis.

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Key Concepts

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

Linear Equation
A linear equation is an equation that models a straight line when it is graphed on a coordinate plane. In essence, it describes a linear relationship between two variables. The most familiar form of expressing a linear equation is the slope-intercept form, written as \( y = mx + b \). Here, \( y \) and \( x \) are the variables, while \( m \) and \( b \) are constants. Each point \((x, y)\) on the line satisfies this equation.
  • Linearity: Linear equations produce straight lines because there is a constant rate of change between the two variables \( x \) and \( y \). This constant rate is known as the slope.
  • Simplicity: The slope-intercept form is particularly simple and useful because it quickly reveals important properties of the line without additional calculations.
Recognizing linear equations is crucial as they serve as the foundation for more complex mathematical topics. When you see \( y = mx + b \), it's important to identify it as the slope-intercept form so you can easily extract the slope and y-intercept.
Slope
The slope of a line, denoted by \( m \) in the slope-intercept form \( y = mx + b \), is a measure of the line's steepness and direction. It is defined as the ratio of the vertical change (change in \( y \)) to the horizontal change (change in \( x \)) between two distinct points on the line.
  • Positive Slope: A line with a positive slope rises from left to right. This means that as \( x \) increases, \( y \) also increases.
  • Negative Slope: A line with a negative slope falls from left to right, indicating that as \( x \) increases, \( y \) decreases.
  • Zero Slope: If the slope is zero, the line is horizontal, showing no change in \( y \) as \( x \) changes.
  • Undefined Slope: A vertical line has an undefined slope because there is no horizontal change, meaning a division by zero would occur when calculating it.
Understanding the slope is essential in predicting how changes in \( x \) affect \( y \) and vice versa. It's the heartbeat of the line's equation, indicating how steep it is and in which direction it tilts.
Y-Intercept
The y-intercept of a line, denoted by \( b \) in slope-intercept form \( y = mx + b \), is the point where the line crosses the y-axis. This point gives a lot of information about the line and its relationship with the coordinate plane:
  • Where: The y-intercept is always at \( x = 0 \), which means it is the value of \( y \) when \( x \) equals zero.
  • Visual Assistance: On a graph, this point helps us quickly locate the starting position of the line at the y-axis, complementing the information given by the slope.
  • Constant Term: In the equation \( y = mx + b \), \( b \) serves as the constant term, reinforcing its role in defining the position of the line vertically on the graph.
Knowing the y-intercept is crucial. It serves as an anchor, a fixed point from where the line extends based on the slope \( m \). Together, the slope and y-intercept give a complete picture of the line's equation in the slope-intercept form.

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Most popular questions from this chapter

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