Problem 91 - Elementary Algebra

Question

Identify the slope of the line represented by $$ y=\left(\frac{40 \mathrm{mi}}{1 \mathrm{hr}}\right) x $$

Step-by-Step Solution

Answer

The slope is \( \frac{40 \text{ mi}}{1 \text{ hr}} \).

Step 1: Identify the Linear Equation Form

The given equation is presented in the form of a linear equation: \[ y = \frac{40 \text{ mi}}{1 \text{ hr}} \times x \]This resembles the general slope-intercept form of a linear equation, which is \[ y = mx + b \].

Step 2: Identify the Slope

In the slope-intercept form, \( y = mx + b \), the coefficient of \( x \) is the slope \( m \). In the given equation, \( y = \frac{40 \text{ mi}}{1 \text{ hr}} \times x \), the coefficient of \( x \) is \( \frac{40 \text{ mi}}{1 \text{ hr}} \).

Key Concepts

Linear Equations

In algebra, a linear equation is an equation that models a straight line when graphed on a coordinate plane. These equations are usually written in the form \[ ax + by = c \], where \( a \), \( b \), and \( c \) are constants. However, one of the most common forms you'll encounter is the slope-intercept form.
Linear equations are essential because they describe relationships with a constant rate of change. Whether you are tracking how much money you spend on groceries over time, or predicting the speed of a car, linear equations can help.

Slope-Intercept Form

The slope-intercept form of a linear equation is useful for quickly identifying the slope and the y-intercept of the line. This form is written as: \[ y = mx + b \], where:

• \( y \) is the dependent variable
• \( m \) is the slope of the line
• \( x \) is the independent variable
• \( b \) is the y-intercept, or the point where the line crosses the y-axis

In the given exercise, the equation \( y = \frac{40 \text{ mi}}{1 \text{ hr}} \times x \) matches this form, where \( \frac{40 \text{ mi}}{1 \text{ hr}} \) is our slope \( m \).

Coefficient of x

In the slope-intercept form of a linear equation, \( y = mx + b \), the term \( m \) is known as the coefficient of \( x \). This coefficient is important because it represents the slope of the line, or how much \( y \) changes for a unit change in \( x \).
In our given problem, \( y = \frac{40 \text{ mi}}{1 \text{ hr}} \times x \), the coefficient of \( x \) is \( \frac{40 \text{ mi}}{1 \text{ hr}} \). This tells us that for every hour that passes, the distance (in miles) changes by 40. The coefficient directly tells you how steep or flat the line will be when graphed. A larger coefficient means a steeper slope, while a smaller coefficient means a gentler slope.