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Chapter 2: Problem 4

Construct a linear equation for each of the following conditions. a. A negative slope and a positive \(y\) -intercept b. A positive slope and a vertical intercept of -10.3 c. A constant rate of change of \(\$ 1300 /\) year

Short Answer

Expert verified
a. \( y = -2x + 5 \)b. \( y = 3x - 10.3 \)c. \( y = 1300x \)

Step by step solution

01

Understanding the Conditions

First, identify the requirements for each linear equation:a. A negative slope and a positive y-intercept.b. A positive slope and a vertical intercept of -10.3.c. A constant rate of change of \( \$1300 / \text{year} \).
02

Creating Equation for Condition a

A linear equation is in the form \( y = mx + b \). For condition (a), the slope \(m\) is negative, and the y-intercept \(b\) is positive. Choose a generic negative slope, e.g., -2, and a positive intercept, e.g., 5:Equation: \( y = -2x + 5 \).
03

Creating Equation for Condition b

For condition (b), the slope \(m\) is positive, and the y-intercept \(b\) is -10.3. Choose a generic positive slope, e.g., 3:Equation: \( y = 3x - 10.3 \).
04

Creating Equation for Condition c

Condition (c) involves a constant rate of change of \( \$1300 / \text{year} \). This rate is the slope \(m\). To construct an equation, assume any y-intercept, e.g., 0 (which means the line passes through the origin):Equation: \( y = 1300x \).

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Key Concepts

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

Understanding Negative Slope
A slope in a linear equation indicates how steep the line is and in which direction it tilts. A negative slope means the line goes downward from left to right. This suggests that as the value of the independent variable (x) increases, the value of the dependent variable (y) decreases.

Usually, the slope is represented by the letter 'm' in the equation format: \( y = mx + b \). If \( m < 0 \), the slope is negative. For instance, if the equation is \({ y = -2x+5 }\), the 'm' value is -2, which is negative, and this indicates a downward trend.

When you see a line with a negative slope, think about it like going downhill on a ski slope. The steeper the slope, the more rapidly 'y' changes as 'x' changes.
Understanding Positive Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In mathematical terms, it is represented by 'b' in the equation \( y = mx + b \). The y-intercept signifies the value of 'y' when 'x' is zero.

A positive y-intercept means the line crosses the y-axis above the origin (0,0). For example, in the linear equation \({ y = -2x + 5 }\), the y-intercept 'b' is 5, which is positive.

To visualize this, imagine you are standing at the origin point on a graph, and the line passes through the point (0, 5) on the y-axis. It starts from there and proceeds according to the slope, which could be positive or negative.

This is an important concept because the y-intercept tells us at what point the line will start plotting on the y-axis before considering the effect of the slope.
Constant Rate of Change
In linear equations, the constant rate of change is essentially the slope of the line. It describes how much 'y' changes for a unit change in 'x'. This concept is particularly useful in real-world problems, such as tracking speed, cost per unit, or any steady growth.

For instance, in the equation \({ y=1300x }\), the slope, which represents the constant rate of change, is 1300. This means for every unit increase in 'x', 'y' increases by 1300. If we think of this in a real-world scenario like salary over years, it means a person earns $1300 each year.

This constant rate provides a straightforward way to analyze and predict behaviors over time. No matter where we start on the graph, the amount of change remains the same, making it very predictable and easy to understand.

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Most popular questions from this chapter

a. Find the value of \(x\) so that the slope of the line through \((x, 5)\) and (4,2) is \(\frac{1}{3}\) b. Find the value of \(y\) so that the slope of the line through (1,-3) and \((-4, y)\) is -2 c. Find the value of \(y\) so that the slope of the line through (-2,3) and \((5, y)\) is 0 d. Find the value of \(x\) so that the slope of the line through (-2,2) and \((x, 10)\) is 2 e. Find the value of \(y\) so that the slope of the line through (-100,10) and \((0, y)\) is \(-\frac{1}{10}\) f. Find at least one set of values for \(x\) and \(y\) so that the slope of line through (5,8) and \((x, y)\) is 0 .

The following table shows U.S. first-class stamp prices (per ounce) over time. $$ \begin{array}{lc} \hline \text { Year } & \text { Price for First-Class Stamp } \\ \hline 2001 & 34 \text { cents } \\ 2002 & 37 \text { cents } \\ 2006 & 39 \text { cents } \\ \hline \end{array} $$ a. Construct a step function describing stamp prices for \(2001-2006 .\) b. Graph the function. Be sure to specify whether each of the endpoints is included or excluded. c. In 2007 the price of a first-class stamp was raised to 41 cents. How would the function domain and the graph change?

Determine which of the following tables represents a linear function. If it is linear, write the equation for the linear function. a. $$ \begin{array}{rr} \hline x & y \\ \hline 0 & 3 \\ 1 & 8 \\ 2 & 13 \\ 3 & 18 \\ 4 & 23 \\ \hline \end{array} $$ b.$$ \begin{array}{rr} \hline q & R \\ \hline 0 & 0.0 \\ 1 & 2.5 \\ 2 & 5.0 \\ 3 & 7.5 \\ 4 & 10.0 \\ \hline \end{array} $$ c. $$ \begin{array}{rr} \hline x & g(x) \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \\ 4 & 16 \\ \hline \end{array} $$ d.$$ \begin{array}{cc} \hline t & r \\ \hline 10 & 5.00 \\ 20 & 2.50 \\ 30 & 1.67 \\ 40 & 1.25 \\ 50 & 1.00 \\ \hline \end{array} $$ e. $$ \begin{array}{rr} \hline x & \multicolumn{1}{c} {h(x)} \\ \hline 20 & 20 \\ 40 & -60 \\ 60 & -140 \\ 80 & -220 \\ 100 & -300 \\ \hline \end{array} $$ f. $$ \begin{array}{cc} \hline p & T \\ \hline 5 & 0.25 \\ 10 & 0.50 \\ 15 & 0.75 \\ 20 & 1.00 \\ 25 & 1.25 \\ \hline \end{array} $$

The accompanying data show rounded average values for blood alcohol concentration \((\mathrm{BAC})\) for people of different weights, according to how many drinks ( 5 oz wine, 1.25 oz 80 -proof liquor, or 12 oz beer) they have consumed. $$ \begin{array}{cccc} \hline \text { Number of Drinks } & 100 \mathrm{lb} & 140 \mathrm{lb} & 180 \mathrm{lb} \\ \hline 2 & 0.075 & 0.054 & 0.042 \\ 4 & 0.150 & 0.107 & 0.083 \\ 6 & 0.225 & 0.161 & 0.125 \\ 8 & 0.300 & 0.214 & 0.167 \\ 10 & 0.375 & 0.268 & 0.208 \\ \hline \end{array} $$ a. Examine the data on BAC for a 100 -pound person. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of the number of drinks, \(D,\) for a 100 -pound person. b. Examine the data on BAC for a 140 -pound person. Are the data linear? If they're not precisely linear, what might be a reasonable estimate for the average rate of change of blood alcohol concentration, \(A,\) with respect to number of drinks, \(D ?\) Find a formula to estimate blood alcohol concentration, \(A,\) as a function of number of drinks, \(D,\) for a 140 -pound person. Can you make any general conclusions about BAC as a function of number of drinks for all of the weight categories? c. Examine the data on \(\mathrm{BAC}\) for people who consume two drinks. Are the data linear? If so, find a formula to express blood alcohol concentration, \(A,\) as a function of weight, \(W,\) for people who consume two drinks. Can you make any general conclusions about \(\mathrm{BAC}\) as a function of weight for any particular number of drinks?

Plot each pair of points, then determine the equation of the line that goes through the points. a. (2,3),(4,0) b. (-2,3),(2,1) c. (2,0),(0,2) d. (4,2),(-5,2)
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