/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Each table describes a linear re... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

Each table describes a linear relationship. For each relationship, find the slope of the line and the \(y\) -intercept. Then write an equation for the relationship in the form \(y=m x+b .\) $$\begin{array}{|c|c|c|c|c|c|}\hline x & {9} & {7} & {5} & {3} & {1} \\\ \hline y & {5} & {4} & {3} & {2} & {1} \\ \hline\end{array}$$

Short Answer

Expert verified
Slope = 0.5, y-intercept = 0.5, Equation: y = 0.5x + 0.5

Step by step solution

01

- Determine the slope (m)

The slope of a linear relationship is calculated using the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Take any two points from the table. Let's use points (9,5) and (7,4). So, \[ m = \frac{4 - 5}{7 - 9} = \frac{-1}{-2} = 0.5 \]
02

- Identify the y-intercept (b)

The y-intercept (b) is the value of y when x is 0. Since the table doesn't provide a point where x is 0, we can use the slope-intercept formula \[ y = mx + b \] Use one of the points from the table, for example, (9,5): \[ 5 = 0.5(9) + b \] Simplify to find b: \[ 5 = 4.5 + b \] \[ b = 5 - 4.5 = 0.5 \]
03

- Write the equation

Now that we have both the slope (m) and the y-intercept (b), we can write the equation of the line in the form \[ y = mx + b \] So, the equation is \[ y = 0.5x + 0.5 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope
The slope of a line describes how steep the line is and the direction it travels. It tells us how much the y-value (vertical position) changes for a given change in the x-value (horizontal position). The slope is often represented by the letter 'm' in equations.
To find the slope, you can use the formula:

It鈥檚 important to understand that the slope is a ratio of change.
  • If the slope is positive, the line goes uphill from left to right.

  • If the slope is negative, the line goes downhill from left to right.

  • If the slope is zero, the line is flat.

  • If the slope is undefined, the line is vertical.
Y-Intercept
The y-intercept of a line is the y-value where the line crosses the y-axis. This is represented by the letter 'b' in equations.
The y-intercept gives us a starting point of the line when x is zero. Knowing the y-intercept along with the slope, we can describe the entire linear relationship.
If you don鈥檛 have a point where x is zero, you can use another known point and rearrange the slope-intercept formula to solve for 'b'. In our case, when we rearranged the equation with one of the points (9,5), we used 0.5 as our slope and calculated 'b' as follows:
[5 = 0.5(9) + b]
Solving this equation gave us the y-intercept 'b' as 0.5.
Linear Relationship
A linear relationship means that there is a consistent rate of change between the variables x and y. In simpler terms, as x changes, y changes at a constant rate. This can be visualized as a straight line on a graph.
Linear relationships are common in many real-world situations, such as calculating costs, predicting growth, and so on.
You can recognize a linear relationship from different representations:
  • Tables: If the rate of change (slope) between x and y is constant.

  • Graphs: If the points form a straight line.

  • Equations: If it can be written in the form y = mx + b.
Slope-Intercept Form
The slope-intercept form of a linear equation is expressed as y = mx + b. This form is extremely useful because it provides immediate insights into the properties of the line:
  • Slope (m): Indicates how steep the line is.

  • Y-intercept (b): Indicates the point where the line crosses the y-axis.

To use the slope-intercept form, follow these steps:
  1. First, find the slope (m).

  2. Next, identify or calculate the y-intercept (b).

  3. Finally, combine both into the equation y = mx + b.
With these elements, you can easily draw the line and understand its behavior.
For example, the equation we derived from the table, y = 0.5x + 0.5, tells us the slope is 0.5 and the line crosses the y-axis at 0.5.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Carlos and Shondra were designing posters for the school play. During the first two days, they created 40 posters. By the third day, they had established a routine, and they calculated that together they would produce 20 posters an hour. a. Make a table like the following that shows how many posters Carlos and Shondra make as they work through the third day. b. Draw a graph to represent the number of posters Carlos and Shondra will make as they work through the third day. c. Write an equation to represent the number of posters they will make as they work through the third day. d. Make a table to show the total number of posters they will have as they work through the third day. e. Draw a graph to show the total number of posters Carlos and Shondra will have as they work through the third day. f. Write an equation that will allow you to calculate the total number of posters they will have based on the number of hours they work. g. Explain how describing just the number of posters created the third day is different from describing the total number of posters created. Is direct variation involved? How are these differences represented in the tables, the graphs, and the equations?

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(-2,\) slope \(-\frac{1}{2}\)

For each set of equations, tell what the graphs of all four relationships have in common without drawing the graphs. Explain your answers. $$\begin{array}{l}{y=1-x} \\ {y=1-2 x} \\ {y=1-3 x} \\ {y=1-4 x}\end{array}$$

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=-2 x $$

Use the distributive property to rewrite each expression without using parentheses. 2\(a\left(0.5 z+z^{2}\right)\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.