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Chapter 1: Problem 27

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=-x $$

Short Answer

Expert verified
a: -1, b: 2

Step by step solution

01

Identify the linear equation

The given linear equation is \( y = -x \). This means for every unit increase in \( x \), \( y \) changes by the coefficient of \( x \), which is -1.
02

Find the constant difference for part a

To find the constant difference between the \( y \) values as the \( x \) values increase by 1, substitute \( x \) with \( x + 1 \). The original equation is \( y = -x \). When \( x \) increases by 1: \( y_1 = -(x+1) = -x - 1 \). The original \( y \) value was \( -x \). Thus, the constant difference is: \( y_1 - y = (-x - 1) - (-x) = -1 \)
03

Find the constant difference for part b

To find the constant difference between the \( y \) values as the \( x \) values decrease by 2, substitute \( x \) with \( x - 2 \). The original equation is \( y = -x \). When \( x \) decreases by 2: \( y_2 = -(x-2) = -x + 2 \). The original \( y \) value was \( -x \). Thus, the constant difference is: \( y_2 - y = (-x + 2) - (-x) = 2 \)

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Key Concepts

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

constant difference
In the context of linear equations, the 'constant difference' is a key concept that refers to the consistent change in the value of the dependent variable (\(y\)) in response to a uniform change in the independent variable (\(x\)). This is a hallmark of linear relationships.
For example, consider the equation given in the exercise: \( y = -x \).
Here:
  • When the value of \(x\) increases by 1, the value of \(y\) changes by the coefficient of \(x\), which is -1. This consistent change of -1 is the 'constant difference'.
  • Likewise, when the value of \(x\) decreases by 2, the value of \(y\) changes by 2 because of the negative sign before \(x\). Thus, the 'constant difference' for a decrease of 2 in \(x\) is +2.
This concept helps in understanding the steady rate at which the values change in a linear equation.
linear relationships
A linear relationship is one in which there is a direct proportionality between the independent variable (\(x\)) and the dependent variable (\(y\)). This means that as one variable changes, the other variable changes at a constant rate.
In simpler terms, if you were to plot this relationship on a graph, it would form a straight line.
For instance, in the equation: \( y = -x \), the changes in \(x\) lead to proportional changes in \(y\).
The characteristics of a linear relationship are:
  • A constant rate of change or slope, which is the 'constant difference.'
  • A straight-line graph representation.
Understanding linear relationships is crucial for solving and graphing linear equations.
slope-intercept form
The slope-intercept form is a way of writing linear equations, and it is written as \( y = mx + b \) where:
  • \(m\) represents the slope.
  • \(b\) represents the y-intercept, or the point where the line crosses the y-axis.
For example, in the exercise, the equation is \( y = -x \). Here:
  • The slope \(m\) is -1, meaning for every 1 unit increase in \(x\), \(y\) decreases by 1 unit.
  • There is no constant term, so the y-intercept \(b\) is 0.
The slope-intercept form is convenient because it immediately tells you the slope and y-intercept of the linear equation, making it easier to graph and understand the relationship between the variables.

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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?

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=2 x+0.25 $$

Make up an equation for a linear relationship. a. Construct a table of \((x, y)\) values for your equation. Include at least five pairs of coordinates. b. Draw a graph using the values in your table are correct. line, check that the values in your table are correct. c. Choose two points from the table, and use them to determine the slope of the line. Check the slope using two other \((x, y)\) pairs.

For Exercises 20鈥�28, answer Parts a and b. a. What is the constant difference between the \(y\) values as the \(x\) values increase by 1\(?\) b. What is the constant difference between the \(y\) values as the \(x\) values decrease by 2\(?\) $$ y=5 x $$

Find an equation of the line passing through the given points. $$ (2,7) \text { and }(6,6) $$
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