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Chapter 2: Problem 5

Determine whether the equation defines \(y\) as a linear function of \(x .\) If so, write it in the form \(y=m x+b\). \(2 x-4 y+9=0\)

Short Answer

Expert verified
The given equation is a linear function of x since the highest power of x is 1. The equation in the form y = mx + b is \(y = \frac{1}{2}x - \frac{9}{4}\), with a slope (m) of \(\frac{1}{2}\) and y-intercept (b) of \(-\frac{9}{4}\).

Step by step solution

01

Isolate y on one side of the equation

Given equation: \(2x - 4y + 9 = 0\) To isolate y, we first move the other terms involving x and the constant to the other side of the equation: Add 4y to both sides: \(2x - 4y + 4y + 9 = 4y\) Subtract 9 from both sides: \(2x - 9 = 4y\) Now we'll divide both sides by 4 to get y by itself: \(\frac{2x - 9}{4} = y\)
02

Check if the equation is linear

A linear equation is an equation where the highest power of the variable x is 1. In our equation, the power of x is 1: \(\frac{2x^1 - 9}{4} = y\) Since the highest power of x is 1, this is a linear equation.
03

Write the equation in the form y = mx + b

Our equation is already in this form: \(y = \frac{2}{4} x - \frac{9}{4}\) So we can simplify it as: \(y = \frac{1}{2} x - \frac{9}{4}\) Our final linear equation is: \(y = \frac{1}{2} x - \frac{9}{4}\) The slope (m) is \(\frac{1}{2}\) and the y-intercept (b) is \(-\frac{9}{4}\).

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is a crucial tool for understanding linear relationships in algebra. It is written as \( y = mx + b \), where \( m \) stands for the slope and \( b \) represents the y-intercept. This form allows you to quickly see how one variable affects another in a linear fashion.

When given an equation that is not in the slope-intercept form, like \( 2x - 4y + 9 = 0 \), your goal is to isolate \( y \) on one side. By rearranging the terms and solving for \( y \), you put the equation into the handy \( y = mx + b \) format. With the equation \( y = \frac{1}{2} x - \frac{9}{4} \), for example, we can immediately identify the slope and y-intercept without further calculation.

By reshaping an equation into this form, you clearly display the direct relationship of \( x \) to \( y \), making it an essential method in both mathematics and its applications.
Slope of a Line
The slope of a line in the slope-intercept form \( y = mx + b \) is represented by \( m \). The slope indicates how steep the line is and the direction it goes, whether up or down. A positive slope means that as \( x \) increases, \( y \) also increases, creating an upward trend.

For example, in the equation \( y = \frac{1}{2} x - \frac{9}{4} \), the slope \( m \) is \( \frac{1}{2} \). This tells us that for every unit we move right along the x-axis, the y-value increases by half a unit. Hence, the line slants upward moderately.

Understanding the slope is essential:
  • If the slope is zero, the line is horizontal.
  • If the slope is positive, the line ascends as it moves from left to right.
  • If the slope is negative, the line descends.
Grasping the idea of slope helps in predicting trends and changes in various scenarios.
Y-Intercept
In the slope-intercept form equation \( y = mx + b \), the y-intercept is the \( b \) term. The y-intercept gives the point where the line crosses the y-axis. It is the value of \( y \) when \( x \) is zero.

For the equation \( y = \frac{1}{2} x - \frac{9}{4} \), the y-intercept is \( -\frac{9}{4} \). This means when \( x = 0 \), \( y \) will be \( -2.25 \). You can visualize this as the starting point or anchor of the line on the vertical axis.

Some important points about y-intercepts include:
  • If \( b \) is positive, the line crosses above the origin.
  • If \( b \) is negative, it crosses below the origin.
  • If \( b \) is zero, the line crosses exactly at the origin.
Knowing the y-intercept helps quickly determine the initial value or starting point of the linear equation's graph.

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Most popular questions from this chapter

The immigration to the United States from Europe, as a percentage of the total immigration, is approximately \(P(t)=0.767 t^{3}-0.636 t^{2}-19.17 t+52.7 \quad(0 \leq t \leq 4)\) where \(t\) is measured in decades, with \(t=0\) corresponding to the decade of the \(1950 \mathrm{~s}\). a. Complete the table: b. Use the result of part (a) to sketch the graph of \(P\). c. Use the result of part (b) to estimate the decade when the immigration, as a percentage of the total immigration, was the greatest and the smallest.

The owner of a luxury motor yacht that sails among the 4000 Greek islands charges \(\$ 600 /\) person \(/\) day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90 ) for the cruise, then each fare is reduced by \(\$ 4\) for each additional passenger. Assume at least 20 people sign up for the cruise and let \(x\) denote the number of passengers above 20 . a. Find a function \(R\) giving the revenue/day realized from the charter. b. What is the revenue/day if 60 people sign up for the cruise? c. What is the revenue/day if 80 people sign up for the cruise?

The relationship between Cunningham Realty's quarterly profit, \(P(x)\), and the amount of money \(x\) spent on advertising per quarter is described by the function $$ P(x)=-\frac{1}{8} x^{2}+7 x+30 \quad(0 \leq x \leq 50) $$ where both \(P(x)\) and \(x\) are measured in thousands of dollars. a. Sketch the graph of \(P\). b. Find the amount of money the company should spend on advertising per quarter in order to maximize its quarterly profits.

Find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=3 x^{2}-5 x+1\)

RISING WATER RATES Based on records from 2001 through 2006, services paid for by households in 60 Boston-area communities that use an average of 90,000 gal of water a year are given by $$ C(t)=2.16 t^{3}+40 t+751.5 \quad(0 \leq t \leq 6) $$ Here \(t=0\) corresponds to 2001 , and \(C(t)\) is measured in dollars/year. What was the average amount paid by a household in 2001 for water and sewer services? If the trend continued, what was the average amount paid in 2008 ?
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