Problem 27 - Elementary Algebra

Question

For Problems \(23-32\), find the equation of the line with the given slope and \(y\) intercept. Leave your answers in slope-intercept form. $$ m=-\frac{1}{6} \text { and } b=-4 $$

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{1}{6}x - 4 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept.

Step 2: Identify Given Values

From the problem, we know that the slope \( m = -\frac{1}{6} \) and the y-intercept \( b = -4 \).

Step 3: Substitute Values into Formula

Substitute the values of \( m \) and \( b \) into the slope-intercept form equation, resulting in \( y = -\frac{1}{6}x - 4 \).

Key Concepts

Linear Equations

A linear equation represents a straight line when it's graphed on a coordinate plane. It is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations have the general form:

• Ax + By = C
• y = mx + b

The first is the standard form, and the second is the slope-intercept form. Linear equations have a degree of one, which means the variable(s) are not squared, cubed, or in any higher power. Because of this feature, their graph is always a straight line.

These equations are essential in algebra and crucial for understanding how real-world data behaves in a linear manner. They describe how two variables relate when one variable depends on another, such as how age can impact income or how distance changes with time.

Slope of a Line

The slope of a line, often represented by the letter \( m \), describes the steepness and the direction of the line. It is calculated as the 'rise over run' between any two points on a line, which can be determined by the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of any two points on the line.

A positive slope means the line is rising from left to right, while a negative slope indicates it is falling. The larger the value of the slope, the steeper the line. A zero slope signifies a perfectly horizontal line, and an undefined slope corresponds to a vertical line.

Understanding the slope is crucial as it helps in analyzing and predicting many real-life situations, such as calculating speed or understanding rates of change.

Y-Intercept

The y-intercept of a line, indicated by \( b \) in the slope-intercept equation \( y = mx + b \), is the point where the line crosses the y-axis. This intercept is the value of \( y \) when \( x = 0 \).

The y-intercept is crucial for determining the starting point of a line on the graph. It helps in sketching the graph quickly because knowing where the line crosses the y-axis allows for easy drawing of the complete line when the slope is also known.

In practical terms, the y-intercept can represent the starting value of a quantity before any changes occur due to another variable. For example, if a graph represents a salary with respect to years of experience, the y-intercept could show the starting salary with no experience.