Problem 17 - Algebra A Combined Function

Question

Mixed Practice Find the slope of each line. See Examples 3 through 6. $$ y=5 x-2 $$

Step-by-Step Solution

Answer

The slope of the line is 5.

Step 1: Identify the linear equation components

The given equation is in the slope-intercept form, which is \( y = mx + b \). In this equation, \( m \) represents the slope, and \( b \) is the y-intercept.

Step 2: Locate the slope in the equation

Compare the given equation \( y = 5x - 2 \) with the standard slope-intercept form \( y = mx + b \). The coefficient of \( x \) is the slope \( m \). Here, \( m = 5 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of expressing the equation of a line. This form is incredibly useful because it allows you to quickly identify critical features of the line, specifically the slope and the y-intercept. The standard format for the slope-intercept form is

• \[ y = mx + b \]

Here, \( m \) denotes the slope of the line, while \( b \) represents the y-intercept, the point where the line crosses the y-axis. The y-intercept is particularly important because it provides a starting point for graphing the line on a coordinate plane. To use the slope-intercept form effectively, match the given linear equation to this format to easily read off the slope and y-intercept. In the original exercise, the equation \( y = 5x - 2 \) fits neatly into the slope-intercept form, with 5 as the slope and -2 as the y-intercept. Understanding this form makes it easy to visualize and graph linear equations.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls as we move from left to right across the graph. Numerically, the slope is often described as 'rise over run', which can be written as

• \[ m = \frac{\Delta y}{\Delta x} \]

where \( \Delta y \) is the change in y-coordinates, and \( \Delta x \) is the change in x-coordinates between two points on the line. In the slope-intercept form \( y = mx + b \), the value of \( m \) gives you the slope directly. A positive slope means the line rises as you move to the right, while a negative slope means it falls. If the slope is zero, the line is horizontal, and in the case of an undefined slope, the line is vertical. In the equation \( y = 5x - 2 \), the slope, \( m \), is 5, suggesting a fairly steep line that rises quickly as we move along the x-axis. Understanding the concept of slope is crucial for interpreting and graphing linear functions.

Linear Functions

Linear functions are mathematical expressions that create a straight line when plotted on a graph. They are characterized by constant changes, meaning that they model situations with a constant rate of increase or decrease. Such functions have a unique property: their graph forms a straight line, which is why they are called 'linear'.The general equation for a linear function is often expressed in the slope-intercept form:

• \[ y = mx + b \]

Important characteristics of a linear function include:

• Constant slope \( m \), which indicates the rate at which the function value changes.
• The y-intercept \( b \), which shows where the line crosses the y-axis.

Linear functions are widely used because they are easy to understand and can model simple relationships between two quantities. In the original exercise, the function \( y = 5x - 2 \) is linear, with a slope of 5, indicating a constant rate of change. Linear functions like this are foundational concepts in algebra and help in developing an understanding of more complex mathematical ideas.