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Chapter 3: Problem 13

Find the slope and the -intercept of the line with the given equation. See Example 1 $$ y=-5 x-8 $$

Short Answer

Expert verified
Slope: -5, Y-intercept: -8.

Step by step solution

01

Identify the Equation Format

The given equation is in the standard slope-intercept form, which is written as \(y = mx + b\). In this equation, \(m\) represents the slope, and \(b\) is the y-intercept.
02

Extract the Slope

In the equation \(y = -5x - 8\), the coefficient of \(x\) is \(-5\). This value represents the slope \(m\). Thus, the slope is \(-5\).
03

Extract the Y-Intercept

The constant term in the equation \(y = -5x - 8\) is \(-8\). This is the value of \(b\), which represents the y-intercept. Therefore, the y-intercept is \(-8\).

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Key Concepts

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

Slope
The slope of a line is a measure of its steepness, indicating how much the line rises or falls as it moves from left to right. In a linear equation written in the slope-intercept form, it is denoted by the letter \(m\). The slope can be positive, negative, zero, or undefined depending on how the line angles in the coordinate plane.
  • **Positive slope:** The line rises as it moves from left to right, indicating a positive change.
  • **Negative slope:** The line falls as it moves from left to right, indicating a negative change, such as in our example \(y = -5x - 8\), where the slope \(m\) is \(-5\).
  • **Zero slope:** A horizontal line has a zero slope because there is no rise as you move along the line.
  • **Undefined slope:** A vertical line has an undefined slope because it does not move horizontally at all.
Understanding the slope helps us know whether the relationship between the variables is increasing or decreasing, and by how much for every one-unit change in the x-direction.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis on a graph. This is an important point for understanding the initial value or starting position of the line when \(x = 0\). In the slope-intercept form of a linear equation \(y = mx + b\), the y-intercept is represented by \(b\).
In the given equation \(y = -5x - 8\), the y-intercept \(b\) is \(-8\). This means that when \(x\) is 0, the value of \(y\) is \(-8\).
Some key points about y-intercepts include:
  • It provides a starting point for graphing the line on a graph.
  • It tells you the value of \(y\) at the exact moment where \(x = 0\).
  • In real-world situations, it often represents a fixed starting value before any changes due to slope are accounted for.
By identifying the y-intercept, you can more easily sketch the graph of the line and analyze its behavior.
Linear Equation
A linear equation is any equation that represents a straight line when plotted on a graph. These equations are fundamental in algebra and often serve as a base for more complex mathematical concepts. A typical linear equation is written in the slope-intercept form, \(y = mx + b\).
Here's why linear equations are crucial:
  • **Predictability:** They allow for easy prediction of one variable based on the changes in another.
  • **Simplicity:** Since they are straight lines, they are easier to work with compared to more complex curves.
  • **Versatility:** Used across various fields like economics, biology, and physics to model relationships.
In the equation \(y = -5x - 8\), you can see:
  • The slope \(m = -5\), telling us the line decreases by 5 units in \(y\) for every 1 unit increase in \(x\).
  • The y-intercept \(b = -8\), showing the line crosses the y-axis at \(-8\).
Linear equations are a vital tool for understanding and modeling real-world relationships in a straightforward and accessible way.

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Most popular questions from this chapter

Find an equation of the line that passes through \((2,5)\) and is parallel to the line \(y=4 x-7 .\) Write the equation in slope-intercept form.

In the function \(y=-5 x+2,\) why do you think the value of \(x\) is called the input and the corresponding value of \(y\) the output?

Streets. One of the steepest streets in the United States is Eldred Street in Highland Park, California (near Los Angeles). It rises approximately 220 feet over a horizontal distance of 665 feet. What is the grade of the street? (IMAGE CANNOT COPY)

Determine whether the lines through each pair of points are parallel, perpendicular, or neither. See Example 8. \((4,2)\) and \((5,-3)\) \((-5,3)\) and \((-2,9)\)

Explain the steps involved in writing \(y-6=4(x-1)\) in slope-intercept form.
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