Problem 7 - Algebra 1: Concepts and Skills

Question

Does the equation model direct variation, inverse variation, or neither? $$ y=7 x-2 $$

Step-by-Step Solution

Answer

The given equation does not model direct variation or inverse variation; instead, it models a linear relation.

Step 1: Identify the form of the given equation

Our given equation is of the form y = 7x - 2, which mirrors the form of the linear equation y = mx + b. Here, m = 7 (slope) and b = -2 (y-intercept).

Step 2: Compare with forms of direct and inverse variation

Direct variation takes the form y = kx or y = x/k, where k is a constant, while inverse variation takes the form y = k/x. Clearly, our given equation does not fit any of these forms.

Step 3: Determine the type of variation

Based on the above comparisons, it is clear that the given equation does not model direct variation or inverse variation. The equation represents a linear equation, not a variation.

Key Concepts

Inverse Variation

Inverse variation describes a relationship between two variables in which the product remains constant. This means that as one variable increases, the other decreases in such a way that their product remains unchanged. The mathematical representation of inverse variation is typically given by the equations

• \( y = \frac{k}{x} \)
• or \( xy = k \)

where \( k \) is a non-zero constant. To identify inverse variation, look for equations where multiplying the variables results in a constant.
Inverse relationships often arise in physics and other sciences. For example, the intensity of light may vary inversely with the square of the distance from the source.
It's crucial to note that inverse variation is different from direct variation. In direct variation, both variables increase or decrease together, whereas in inverse variation, one increases while the other decreases.

Linear Equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations take the form

• \( y = mx + b \)

where \( m \) represents the slope of the line and \( b \) is the y-intercept. The slope, \( m \), indicates the steepness of the line, while the y-intercept, \( b \), is the point where the line crosses the y-axis.
Linear equations graph as straight lines on a Cartesian plane. They model relationships that have a constant rate of change.
For example, if you earn $7 per hour, the equation \( y = 7x \) can represent your total earnings \( y \) for \( x \) hours worked.

• This equation has a slope of 7, representing the "per hour" rate.
• If a linear equation has a y-intercept, it means the line doesn't pass through the origin (0,0).

Linear equations are foundational in algebra and are used extensively in both academic and real-world applications.

Slope-Intercept Form

The slope-intercept form of a linear equation provides a straightforward way to understand the line's properties. This form is expressed as:

• \( y = mx + b \)

where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) measures how steep the line is. It is calculated as the change in \( y \) over the change in \( x \) between two points on the line.
The y-intercept \( b \) is the point at which the line crosses the y-axis. This is where \( x = 0 \).

• For instance, in the equation \( y = 7x - 2 \), \( m = 7 \) and \( b = -2 \).
• The slope \( 7 \) indicates that for every unit increase in \( x \), \( y \) increases by 7.
• The y-intercept \( -2 \) tells us that the line crosses the y-axis at (0, -2).

Using the slope-intercept form makes graphing linear equations and understanding their behavior relatively simple. It also helps in quickly identifying the rate of change and initial value represented by the line.