Problem 14 - Algebra A Combined Function

Question

Write an equation of the line with each given slope, \(m\), and \(y\) -intercept, \((0, b) .\) $$ m=-3, b=-3 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

You need to write the equation of a line using the given slope and the y-intercept. The slope, denoted by \( m \), is given as \(-3\) and the y-intercept, denoted by \( b \), is given as \(-3\).

Step 2: Use the Slope-Intercept Form

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

Step 3: Substitute the Slope

Substitute \( m = -3 \) into the equation \( y = mx + b \). This gives us the equation: \( y = -3x + b \).

Step 4: Substitute the Y-intercept

Substitute \( b = -3 \) into the equation. This gives us the final equation for the line: \( y = -3x - 3 \).

Step 5: Write the Final Equation

The equation of the line with a slope of \(-3\) and y-intercept \(-3\) is \( y = -3x - 3 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a fundamental way to express the equation of a straight line. It is written as \[ y = mx + b \]. This form makes it easy to quickly identify both the slope and the y-intercept of the line.

• The letter \( y \) represents the dependent variable in the equation, commonly associated with the vertical axis in a graph.
• The letter \( x \) represents the independent variable, often positioned on the horizontal axis.
• The slope \( m \) determines the line’s steepness and direction, while the \( b \) in the formula is the value where the line crosses the y-axis.

By using the slope-intercept form, creating and understanding the relationship between two variables becomes efficient, especially if you know the slope and y-intercept beforehand.

Slope of a Line

The slope of a line, denoted as \( m \) in the slope-intercept form, is crucial for understanding how steep a line is. It tells us how much the line will rise or fall as we move from left to right on a graph.

• A positive slope means the line rises as it moves to the right.
• A negative slope indicates the line falls to the right, as in our example with \( m = -3 \).
• A zero slope results in a horizontal line, which means there's no rise or fall.
• If the slope is undefined, the line is vertical.

The slope is calculated by the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line. So, the larger the absolute value of \( m \), the steeper the line.

Y-intercept

The y-intercept of a line is the point where it crosses the y-axis. In the slope-intercept form equation \( y = mx + b \), this y-intercept is represented by \( b \).

• The y-intercept is particularly useful because it gives an exact starting point of the line on the vertical axis.
• For the given problem, the y-intercept \( b \) is \(-3\), meaning the line crosses the y-axis at \( (0,-3) \).
• It is the value of \( y \) when \( x = 0 \).

Understanding the y-intercept allows one to easily graph the line, beginning from a known point and using the slope to determine the direction and steepness of the line.

Equation of a Line

Writing the equation of a line involves combining both the slope and the y-intercept into one coherent expression. Using the slope-intercept form \( y = mx + b \):

• Start by substituting the known slope \( m \) into the equation.
• Then, substitute the y-intercept \( b \).
• For this exercise, by inserting \( m = -3 \) and \( b = -3 \), we derived the equation \( y = -3x - 3 \).
• This equation encapsulates both the direction and steepness of the line through the slope and the specific point where the line meets the y-axis.

The final equation provides a complete description of the line, making it easy to graph or use for further calculations.