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Chapter 3: Problem 7

Write an equation of the line with each given slope, \(m,\) and \(y\) -intercept, \((0, b) .\) See Example \(1\). $$ m=0, b=-8 $$

Short Answer

Expert verified
The equation of the line is \( y = -8 \).

Step by step solution

01

Identify the Line Equation Format

The equation for a straight line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Substitute the Slope

Given that \( m = 0 \), we substitute this value into the equation. Thus, the equation becomes \( y = 0 \cdot x + b \).
03

Substitute the Y-intercept

Given that \( b = -8 \), we substitute this into the equation \( y = 0 \cdot x + (-8) \), which simplifies to \( y = -8 \).
04

Simplified Equation

Since the slope is \( 0 \), the term \( 0 \cdot x \) disappears, leaving us with the equation \( y = -8 \). This is the final equation of the line.

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Key Concepts

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

Linear Equations
Linear equations are fundamental in mathematics and represent lines on a graph. They are commonly written in the slope-intercept form as \( y = mx + b \). This form is particularly useful because it clearly indicates how a change in \( x \) affects \( y \).
  • \( y \) stands for the dependent variable or the variable output of the function.
  • \( x \) represents the independent variable or input.
  • \( m \) is the slope of the line, indicating its steepness or incline.
  • \( b \) is the y-intercept, showing where the line crosses the y-axis.
A linear equation graphically shows a straight line. The goal is to determine this line given certain information, like the slope \( m \) and y-intercept \( b \). Understanding how to read and write linear equations is crucial for solving and graphing them effectively.
Y-Intercept
The y-intercept of a line is a critical concept when understanding linear equations. It is the point at which the line crosses the y-axis.
By knowing the y-intercept \( b \), you can quickly determine a key point on the graph without any calculations once the slope-intercept form is used.
For the line equation \( y = mx + b \):
  • The y-intercept \( b \) is the value of \( y \) when \( x = 0 \).
  • This is useful because it allows you to start plotting the line on a graph by marking the intercept on the y-axis.
In our exercise, we have \( b = -8 \), meaning the line crosses the y-axis at \( y = -8 \). Hence, this point (0, -8) becomes a crucial starting point for plotting the line.
Remember, the y-intercept always gives you a fixed point on the graph, simplifying the process of drawing the entire line when combined with the slope information.
Slope
The slope of a line is a measure of its steepness and direction. It tells us how much \( y \) changes for a given change in \( x \). In the slope-intercept form of a linear equation, \( y = mx + b \), \( m \) is the slope.
  • A positive slope means the line is ascending from left to right.
  • A negative slope indicates the line is descending from left to right.
  • A zero slope, like \( m = 0 \) in our exercise, indicates a horizontal line.
      • In our example, with a given slope \( m = 0 \), you will notice that the line is perfectly horizontal. The equation simplifies to \( y = -8 \), showing that for any value of \( x \), \( y \) remains constant at \(-8\).Understanding the impact of the slope helps you predict and visualize how the line behaves on a graph. It also guides you in understanding relationships between variables in real-world situations described by linear equations.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. \(y=\frac{2}{9} x+3\) \(y=-\frac{2}{9} x\)

There were approximately \(14,774\) kidney transplants performed in the United States in \(2002 .\) In \(2006,\) the number of kidney transplants performed in the United States rose to 15. \(722 .\) (Source: Organ Procurement and Transplantation Network) a. Write two ordered pairs of the form (year, number of kidney transplants). b. Find the slope of the line between the two points. c. Write a sentence explaining the meaning of the slope as a rate of change.

In your own words, describe how to plot an ordered pair.

Solve each equation for \(y.\) \(y-7=-9(x-6)\)

Given the equation of a nonvertical line, explain how to find the slope without finding two points on the line.
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