Problem 25 - Precalculus Mathematics for Calculus

Question

Find an equation of the line that satisfies the given conditions. Slope \(3 ; \quad y\) -intercept \(-2\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 3x - 2 \).

Step 1: Recall the Slope-Intercept Form

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

Step 2: Substitute the Given Slope

Substitute the given slope \( m = 3 \) into the slope-intercept formula. This gives us \( y = 3x + b \).

Step 3: Substitute the Given Y-Intercept

Now, substitute the given y-intercept \( b = -2 \) into the equation from Step 2. This yields \( y = 3x - 2 \).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that describe a straight line when graphed. They have a constant rate of change. This means every time the value of the variable changes, the graph's rise to run ratio remains constant. A linear equation can come in different forms, but one commonly used format is the slope-intercept form:

• The standard form is given as: \( Ax + By = C \).
• The slope-intercept form: \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept.
• Each form of a linear equation tells us different things about the line, such as its slope, y-intercept, and direction.

Linear equations are central to algebra and are fundamental to understanding concepts in more advanced mathematics. With these equations, you can model and solve real-world problems that require constant change or growth.

Slope

The slope of a line is a measure of its steepness and direction. It tells us how many units the line rises or falls for a unit increment horizontally. The slope is represented by \( m \) in the equation \( y = mx + b \). It is defined mathematically as:

• \( m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} \), which means the change in \( y \) over the change in \( x \).
• A positive slope means the line rises as it moves from left to right.
• A negative slope indicates the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal, indicating no vertical change.
• A vertical line has an undefined slope because division by zero is not possible.

Understanding the slope is crucial, as it determines how "tilted" a line is on a graph, and provides information about the rate of change between the x and y variables.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In the slope-intercept form equation \( y = mx + b \), \( b \) represents the y-intercept. This point shows the value of \( y \) when \( x = 0 \). Here's what we need to know about the y-intercept:

• It's crucial for graphing a line because it provides a starting point.
• For the given equation from the exercise, the y-intercept is \(-2\), meaning when \( x = 0 \), \( y = -2 \).
• Every line has exactly one y-intercept unless it's vertical, in which case the line may not cross the y-axis.

The y-intercept helps us understand the starting value of a function, and it can be useful in determining where a line is positioned on a graph. Knowing \( b \) allows us to plot the line quickly and understand the behavior of linear functions at the origin.