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Chapter 2: Problem 60

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Slope \(=-2 ; y\) -intercept \(=-2\)

Short Answer

Expert verified
The equation is \ y = -2x - 2 \.

Step by step solution

01

Identify the slope and y-intercept

We are given the slope of the line as \(-2\) and the y-intercept as \(-2\). In the slope-intercept form of the equation of a line, which is \(y = mx + b\), \m\ represents the slope, and \b\ represents the y-intercept.
02

Plug in the values

Substitute \m = -2\ and \b = -2\ into the slope-intercept form \(y = mx + b\). This gives us: \ y = -2x - 2 \.
03

Review the equation

The equation \(y = -2x - 2\) is now in the slope-intercept form. This means we've found the equation of the line with the specified properties.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form of a line is crucial for understanding linear equations quickly. This form is written as: \[ y = mx + b \]In this equation,
  • \( m \) stands for the slope of the line
  • \( b \) represents the y-intercept, which is where the line crosses the y-axis
Using the slope-intercept form makes it simple to graph lines and understand their behavior. Simply plug in values for \( m \) and \( b \) to get the equation of your line.
slope
The slope is a measure of how steep a line is. It is denoted by the letter \( m \) in the slope-intercept form of a line. The slope describes the rate at which \( y \) changes with respect to \( x \). The formula for the slope is given by: \[ m = \frac{rise}{run} \]
  • \( rise \) is the vertical change
  • \( run \) is the horizontal change
In the given problem, the slope is \( -2 \), meaning for every unit increase in \( x \), \( y \) decreases by 2 units. Negative slopes indicate that the line is decreasing.
y-intercept
The y-intercept is where the line crosses the y-axis. It is represented by the letter \( b \) in the slope-intercept formula. To find it, set \( x = 0 \) in the equation and solve for \( y \). In this exercise, the y-intercept provided is \( -2 \), meaning the point \( (0, -2) \) is where the line crosses the y-axis.Understanding the y-intercept helps in quickly graphing the line and knowing one key point through which it passes.

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