$y = mx + b$

Definition

The equation $y = mx + b$ represents a linear function, where $y$ is the dependent variable, $x$ is the independent variable, $m$ is the slope, and $b$ is the $y$-intercept. This equation describes a straight line on a coordinate plane.

5 Must Know Facts For Your Next Test

1. The slope, $m$, can be positive, negative, zero, or undefined, indicating the direction and steepness of the line.
2. The $y$-intercept, $b$, represents the value of $y$ when $x = 0$, and it shifts the line vertically on the coordinate plane.
3. The equation $y = mx + b$ can be used to graph linear equations by plotting points or using the slope-intercept form.
4. The slope-intercept form, $y = mx + b$, is one of the most common ways to represent and work with linear equations.
5. The equation $y = mx + b$ can be used to model and analyze various real-world linear relationships, such as the cost of an item based on the quantity purchased.

Review Questions

• Explain how the slope, $m$, and the $y$-intercept, $b$, affect the graph of the linear equation $y = mx + b$.
  • The slope, $m$, determines the steepness and direction of the line. A positive slope indicates the line is sloping upward, a negative slope indicates the line is sloping downward, a zero slope indicates a horizontal line, and an undefined slope indicates a vertical line. The $y$-intercept, $b$, represents the point where the line crosses the $y$-axis, shifting the line vertically on the coordinate plane. Together, the slope and $y$-intercept define the unique characteristics of the linear equation and its corresponding graph.
• Describe how the equation $y = mx + b$ can be used to model and analyze real-world linear relationships.
  • The equation $y = mx + b$ is commonly used to model and analyze various linear relationships in the real world. For example, it can be used to represent the cost of an item based on the quantity purchased, where $y$ represents the total cost, $x$ represents the quantity, $m$ represents the unit price, and $b$ represents any fixed costs. By using this equation, one can determine the cost for any given quantity, predict the quantity needed to reach a certain cost, or analyze the impact of changes in the unit price or fixed costs on the overall cost.
• Explain how the slope-intercept form, $y = mx + b$, is a useful way to work with and understand linear equations.
  • The slope-intercept form, $y = mx + b$, is a widely used and valuable way to represent and work with linear equations because it clearly identifies the key characteristics of the line: the slope, $m$, and the $y$-intercept, $b$. This form allows for easy graphing of the line, as well as the ability to quickly determine the rate of change (slope) and the starting point (y-intercept) of the linear relationship. Furthermore, the slope-intercept form is commonly used in various applications, such as modeling linear functions in real-world scenarios, analyzing the relationship between variables, and making predictions based on the given information.

Related terms

Slope:

The slope, $m$, represents the rate of change or the steepness of the line. It indicates how much the $y$-value changes for a unit change in the $x$-value.

$y$-intercept: The $y$-intercept, $b$, is the point where the line crosses the $y$-axis, indicating the value of $y$ when $x = 0$.

Linear Equation:

A linear equation is an equation in which the highest exponent of the variable is 1. The graph of a linear equation is a straight line.