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Chapter 1: Problem 6

Determine whether the equation defines \(y\) as a linear function of \(x .\) If so, write it in the form \(y=m x+b\). $$ 3 x-6 y+7=0 $$

Short Answer

Expert verified
The given equation \(3x - 6y + 7 = 0\) is a linear function, as it can be written in the form \(y = mx + b\). The equation in this form is \(y = \frac{1}{2}x - \frac{7}{6}\), with a slope \(m = \frac{1}{2}\) and a y-intercept \(b = -\frac{7}{6}\).

Step by step solution

01

Check if it's a linear function

If the equation can be written in the form y = mx + b, where m and b are constants, then it is a linear function. Let's try to rewrite the given equation in that form.
02

Isolate y from the given equation

First, let's isolate y in the given equation \(3x - 6y + 7 = 0\): $$ 6y = 3x - 7 $$
03

Solve for y

Now divide both sides by 6 to solve for y: $$ y = \frac{1}{2}x - \frac{7}{6} $$
04

Identify m and b

We can see the equation is in the form y = mx + b, where: $$ m = \frac{1}{2} $$ $$ b = - \frac{7}{6} $$ Since we were able to write the equation in the form y = mx + b, it is a linear function. The equation of y as a linear function of x in the form y = mx + b is as follows: $$ y = \frac{1}{2}x - \frac{7}{6} $$

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Key Concepts

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

Equation of a Line
The equation of a line is a mathematical statement showing the relationship between two variables, typically \(x\) and \(y\). It represents a line on a graph. The simplest and most common form of this equation is the slope-intercept form, which is represented as \( y = mx + b \). Here, \( m \) represents the slope of the line, which indicates the line's steepness, while \( b \) is the y-intercept, indicating where the line crosses the y-axis.
Understanding the equation of a line is crucial because it helps in identifying whether a given relationship is linear. When given a complex equation like \( 3x - 6y + 7 = 0 \), we check if we can transform it into the form \( y = mx + b \). If yes, the equation is linear, meaning it describes a straight line.
So, by rewriting the original equation, we isolate \( y \) and determine the slope and y-intercept, thus understanding the line's direction and starting point on the graph.
Slope-Intercept Form
The slope-intercept form is a way of expressing the equation of a line as \( y = mx + b \). This form makes it easy to identify important characteristics of the line:
  • **Slope (\( m \))**: Refers to the line's tilt鈥攚hether it's rising or falling, and how steeply.
  • **Y-Intercept (\( b \))**: Shows the starting point of the line on the y-axis.
To convert an equation into the slope-intercept form, we often need to rearrange and solve the equation to isolate \( y \). For example, with the original equation \( 3x - 6y + 7 = 0 \), rearranging gives \( 6y = 3x - 7 \). Dividing through by 6 simplifies to \( y = \frac{1}{2}x - \frac{7}{6} \).
This rearranged equation clearly shows the slope \( \frac{1}{2} \) and the y-intercept \( -\frac{7}{6} \). Thus, expressing a line in the slope-intercept form provides a straightforward way to understand and graph the line easily.
Solving Linear Equations
Solving linear equations involves finding the value of one variable in terms of another. They may include one or multiple steps such as isolating the variable, rearranging terms, and simplifying, to express the equation in a standard form like \( y = mx + b \).
Consider solving the equation \( 3x - 6y + 7 = 0 \). To solve for \( y \), we aim to isolate it on one side. First, we subtract 3x and 7 from both sides, getting \( 6y = 3x - 7 \). Next, we divide by 6 to solve for \( y \), resulting in \( y = \frac{1}{2}x - \frac{7}{6} \).
This process involved understanding and manipulating terms. It's about performing inverse operations step-by-step to simplify the equation finalizing with an equation showing \( y \) in terms of \( x \), informed by the concepts of equation solving and variable isolation. Solving linear equations is essential in mathematics as it allows us to express relationships between variables clearly.

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Most popular questions from this chapter

The following table gives the projected U.S. online banking households as a percentage of all U.S. banking households from \(2001(x=1)\) through \(2007(x=7)\) : $$ \begin{array}{lccccccc} \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Percentage of } \\ \text { Households, } \boldsymbol{y} \end{array} & 21.2 & 26.7 & 32.2 & 37.7 & 43.2 & 48.7 & 54.2 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the projected percentage of U.S. online banking households in 2008 .

The lines with equations \(a x+b y+c_{1}=0\) and \(b x-a y+\) \(c_{2}=0\), where \(a \neq 0\) and \(b \neq 0\), are perpendicular to each other.

The relationship between the temperature in degrees Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) and the temperature in degrees Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) is $$ F=\frac{9}{5} C+32 $$ a. Sketch the line with the given equation. b. What is the slope of the line? What does it represent? c. What is the \(F\) -intercept of the line? What does it represent?

With computer security always a hot-button issue, demand is growing for technology that authenticates and authorizes computer users. The following table gives the authentication software sales (in billions of dollars) from 1999 through \(2004(x=0\) represents 1999): $$ \begin{array}{ccccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Sales, } \boldsymbol{y} & 2.4 & 2.9 & 3.7 & 4.5 & 5.2 & 6.1 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the sales for 2007 , assuming the trend continues.

The number of U.S. broadband Internet households (in millions) between the beginning of \(2004(t=0)\) and the beginning of \(2008(t=4)\) was estimated to be $$ f(t)=6.5 t+33 \quad(0 \leq t \leq 4) $$ Over the same period, the number of U.S. dial-up Internet households (in millions) was estimated to be $$ g(t)=-3.9 t+42.5 \quad(0 \leq t \leq 4) $$ a. Sketch the graphs of \(f\) and \(g\) on the same set of axes. b. Solve the equation \(f(t)=g(t)\) and interpret your result.
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