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Chapter 2: Problem 8

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ y=3 x-5 $$

Short Answer

Expert verified
Yes, it's linear with slope 3 and y-intercept -5.

Step by step solution

01

Identify the Given Equation

The given equation is \( y = 3x - 5 \). A linear function generally has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
02

Compare to Linear Function Form

The equation \( y = 3x - 5 \) resembles the linear function form \( y = mx + b \). Here, \( m = 3 \) and \( b = -5 \), which fits the structure of a linear function.
03

Conclusion on Linearity

Since the equation \( y = 3x - 5 \) matches the form of a linear function \( y = mx + b \), where both the slope and y-intercept are clearly defined, it can be written as a linear function.

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Key Concepts

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

Equation of a Line
An equation of a line in algebra represents a straight line in a two-dimensional plane. The standard form of a linear equation is given by the formula: \[ y = mx + b \]Here,
  • \(y\) is the dependent variable representing the output of the function.
  • \(x\) is the independent variable representing the input.
  • \(m\) is the slope of the line.
  • \(b\) is the y-intercept, which indicates where the line crosses the y-axis.
Breaking down the equation gives us insight into the behavior and composition of the line. This equation indicates how the value of \(y\) changes in response to variations in \(x\). As \(x\) increases or decreases, the line's slope \(m\) dictates the degree of this change. Meanwhile, the constant \(b\) determines the starting point of the line on the y-axis, crucial for sketching and understanding the line's graph.
Slope
The slope \(m\) of a line is a measure of its steepness and direction. In the linear equation \[ y = mx + b \], \(m\) plays a pivotal role in defining the relationship between \(x\) and \(y\). If we consider two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated by the formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula represents the "rise over run", the change in \(y\) over the change in \(x\), showing how \(y\) increases or decreases with \(x\).
  • A positive slope means the line ascends from left to right.
  • A negative slope implies the line descends.
  • When the slope is zero, the line is horizontal, indicating no change in \(y\) despite variations in \(x\).
  • If the slope is undefined, it indicates a vertical line where \(x\) remains constant.
Interpreting the slope enables us to predict how variables are related and how one reacts to changes in the other.
Y-intercept
The y-intercept \(b\) in the linear equation \[ y = mx + b \]represents the point where the line crosses the y-axis. This crossing happens when the value of \(x\) is zero. Thus, it acts as a starting point for the line on the Cartesian plane.
  • It provides a reference point for graphing the line.
  • It conveys the initial value of \(y\) when \(x = 0\), often useful for real-world interpretations, like initial conditions in physics or economics.
Visualizing the y-intercept helps in understanding the line's placement in the plane and forms the basis for plotting the line efficiently. Together with the slope, the y-intercept brings the full equation to life, making it possible to represent various linear scenarios accurately.

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Most popular questions from this chapter

Find the equation of the line that passes through the following points: \((2 a, b)\) and \((a, b+1)\)

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For the following exercises, consider this scenario: The median home values in subdivisions Pima Central and East Valley (adjusted for inflation) are shown in Table 2.14. Assume that the house values are changing linearly. $$\begin{array}{|c|c|c|}\hline \text { Year } & {\text { Pima Central }} & {\text { East valley }} \\ \hline 1970 & {32,000} & {120,250} \\ \hline 2010 & {85,000} & {150,000} \\ \hline\end{array}$$ In which subdivision have home values increased at a higher rate?

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular: $$-2 x+y=3$$ $$3 x+\frac{3}{2} y=5$$

In \(2003,\) a town's population was \(1431 .\) By 2007 the population had grown to 2134 . Assume the population is changing linearly. a. How much did the population grow between the year 2003 and 2007\(?\) b. How long did it take the population to grow from 1431 people to 2134 people? c. What is the average population growth per year? d. What was the population in the year 2000 ? e. Find an equation for the population, \(P\) of the town \(t\) years after 2000 . f. Using your equation, predict the population of the town in 2014 .
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