Problem 61 - Elementary Algebra

Question

Write the equation of the line using the given information. Write the equation in slope-intercept form. Slope \(=1, \quad y\) -intercept \(=-2\)

Step-by-Step Solution

Answer

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

Step 1: Identify the slope and y-intercept

The given information tells us that the slope (m) is equal to 1 and the y-intercept (b) is equal to -2.

Step 2: Write the equation in slope-intercept form

Plug the given values for m and b into the slope-intercept form equation, which is \(y = mx + b\). In this case, the equation is \(y = 1x - 2\). Since 1 times x is just x, we can simplify the equation to: \(y = x - 2\) This is the equation of the line in slope-intercept form.

Key Concepts

Equation of a Line

When we talk about the equation of a line, we're referring to a mathematical equation that describes how a straight line looks on a graph. One of the most common forms for writing the equation of a line is called the **slope-intercept form**. It's a specific way of expressing the equation that uses parameters to define the line clearly.

The slope-intercept form is given by:

• \( y = mx + b \)

Here, \( y \) and \( x \) are the variables representing points on the line.

• \( m \) represents the slope of the line.
• \( b \) is the y-intercept where the line crosses the y-axis.

This form provides a straightforward way to understand and visualize the line's behavior by focusing on two main attributes – how steep the line is and where it starts on the y-axis.

Slope

The slope of a line is a measure of its steepness or its rate of change. Think of the slope as a way to describe how slanted the line is.

In mathematical terms, the slope \( m \) is defined as:

• The ratio of the "rise" (vertical change) and "run" (horizontal change) between two points on the line.
• This is often expressed as \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the difference in the y-values and \( \Delta x \) is the difference in the x-values between these two points.

For our specific problem, we're given that the slope \( m \) is 1. This means for every unit you move along the x-axis, you move the same amount along the y-axis. Hence, the line increases by one unit up for every unit it moves to the right. A slope of 1 indicates a very balanced and diagonal line moving towards the top-right of the graph.

Y-Intercept

The **y-intercept** is a vital component in understanding the equation of a line. It specifically tells you where the line crosses the y-axis on a graph. The y-intercept is represented by \( b \) in the slope-intercept form \( y = mx + b \).

• In our problem, the y-intercept \( b \) is \(-2\).
• This means the line meets the y-axis at the point \((0, -2)\).

This point is crucial because it provides a starting position for the line. From here, the slope tells us how the line will move across the graph. Knowing both the slope and the y-intercept lets you easily draw the line, predict its path, and understand the relationships represented by the line in practical scenarios.