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Chapter 3: Problem 18

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ y=12-4 x $$

Short Answer

Expert verified
The slope is -4, and the y-intercept is 12.

Step by step solution

01

Identify the equation format

This equation is given in the format \(y = -4x + 12\), which is similar to the standard slope-intercept form of a line equation \(y = mx + b\). Here, \(m\) is the slope, and \(b\) is the y-intercept.
02

Determine the slope \(m\)

In the equation \(y = -4x + 12\), the coefficient of \(x\) is \(-4\). Therefore, the slope \(m\) of the line is \(-4\).
03

Determine the y-intercept \(b\)

In the equation \(y = -4x + 12\), the constant term without \(x\) is \(12\). Therefore, the y-intercept \(b\) is \(12\).

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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 is a way of expressing the equation of a straight line. It's one of the most common and easy-to-understand forms for linear equations. By organizing an equation in this way, you immediately know two crucial features of the line: the slope and the y-intercept. The general formula in slope-intercept form is \( y = mx + b \), where:
  • \( y \) represents the y-coordinate of any point on the line,
  • \( x \) represents the x-coordinate of that point,
  • \( m \) is the slope,
  • \( b \) is the y-intercept.
This format is particularly helpful when you want to quickly graph a line or understand how steep a line is. For instance, the original exercise's equation \( y = 12 - 4x \) can be rearranged as \( y = -4x + 12 \), clearly showcasing the slope and the intercept.
Slope
Slope is a measure of how steep a line is. In the slope-intercept form \( y = mx + b \), the slope is represented by the letter \( m \). It tells you how much \( y \) increases or decreases as \( x \) increases by 1.
  • If \( m \) is positive, the line rises as you move from left to right.
  • If \( m \) is negative, the line falls as you move from left to right.
  • If \( m \) is zero, the line is perfectly horizontal.
In the example equation \( y = -4x + 12 \), the slope \( m \) is \(-4\). This means that for every increase of 1 in \( x \), \( y \) decreases by 4, indicating a downward slope.
Y-Intercept
The y-intercept is a point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). This point is crucial because it's where \( x = 0 \).
  • The y-intercept gives the starting value of \( y \) when \( x \) is zero.
  • It helps in quickly drawing or visualizing the line on a graph.
  • The y-intercept is often where the graphing of the line begins.
In the equation \( y = -4x + 12 \), the y-intercept \( b \) is 12. This tells us that the line will cross the y-axis at the point (0, 12). Knowing this point makes it easier to place other points and sketch the line accurately.

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