Problem 66 - Intermediate Algebra

Question

What is the slope-intercept form of the equation of a line?

Step-by-Step Solution

Answer

The slope-intercept form is \( y = mx + b \).

Step 1: Identify the Formula

The slope-intercept form of a linear equation is a standard way of representing a straight line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept.

Step 2: Understand the Slope \( m \)

The slope \( m \) refers to the steepness and direction of the line. It is calculated as the change in \( y \) divided by the change in \( x \) between two distinct points on the line, often written as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Step 3: Recognize the Y-Intercept \( b \)

The y-intercept \( b \) is the value of \( y \) when the line crosses the y-axis. This occurs when \( x = 0 \).

Step 4: Formulating the Equation

Once you have the slope \( m \) and the y-intercept \( b \), substitute these values into the slope-intercept formula to construct the linear equation: \( y = mx + b \).

Key Concepts

Slope

The concept of slope is fundamental to understanding how lines behave on a graph. In the slope-intercept form, represented as \( y = mx + b \), \( m \) denotes the slope. Think of it as the measure of how steep a line is.

The slope tells us how much \( y \) increases or decreases as \( x \) increases by one unit. This is often described with the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on the line.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• If the slope is zero, the line is horizontal.
• An undefined slope indicates a vertical line.

Y-Intercept

The y-intercept is another key part of the slope-intercept equation. It's a simple, yet crucial concept to grasp: it represents the point where the line crosses the y-axis, which is the point where \( x = 0 \).

In the slope-intercept equation \( y = mx + b \), \( b \) is the y-intercept. Visually, it provides a starting point from which the slope can direct the line.

• If \( b > 0 \), the intercept is above the origin.
• If \( b < 0 \), the intercept is below the origin.
• If \( b = 0 \), the line passes through the origin.

Linear Equation

A linear equation represents a straight line and is one of the simplest forms of algebraic expressions.

The slope-intercept form \( y = mx + b \) is a standardized way of writing a linear equation. Each linear equation has exactly one slope and one y-intercept, making it predictable and easy to graph.

• Linear equations model real-world situations involving constant rates of change.
• They can be used to describe relationships between variables or to predict future values based on existing data.
• Understanding the linear equation helps in solving problems related to distance, time, speed, and more.