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Chapter 3: Problem 2

A line with given vertical intercept and slope: On coordinate axes, draw a line with vertical intercept 3 and slope 1. What is its horizontal intercept?

Short Answer

Expert verified
The horizontal intercept of the line is -3.

Step by step solution

01

Understanding the Line Equation

The equation of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the vertical (y-axis) intercept. For this problem, the slope \( m = 1 \) and the vertical intercept \( b = 3 \). Therefore, the equation of the line is \( y = x + 3 \).
02

Finding the Horizontal Intercept

The horizontal (x-axis) intercept occurs where the line crosses the x-axis, meaning \( y = 0 \). Set \( y = 0 \) in the equation \( y = x + 3 \) to solve for \( x \).
03

Solving for the Horizontal Intercept

Set the equation to zero: \[ 0 = x + 3 \]Subtract 3 from both sides to solve for \( x \):\[ x = -3 \]Thus, the horizontal intercept is \( x = -3 \).

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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 is a simple and powerful way to express the equation of a straight line.This form is written as \( y = mx + b \).
  • Here, \( m \) represents the slope of the line. The slope is a measure of how steep the line is. It indicates how much \( y \) increases or decreases as \( x \) increases by one unit.
  • The \( b \) in the equation is the vertical intercept. This tells us where the line crosses the vertical (y) axis.
In slope-intercept form, you can quickly understand the direction and position of the line just by looking at \( m \) and \( b \).
For example, with our given line equation \( y = x + 3 \), the slope \( m \) is 1. This means for every increase of 1 in \( x \), \( y \) also increases by 1.
The vertical intercept \( b \) is 3, indicating the line starts at the point (0, 3) on the y-axis.
Vertical Intercept
The vertical intercept is a crucial concept in understanding linear equations and how they graphically appear on the coordinate plane.
It is the point where the line crosses the y-axis, represented in the slope-intercept equation \( y = mx + b \) by the \( b \) value.
  • This intercept is important because it gives a starting point for the line, allowing you to trace the line's path as it rises or falls based on the slope.
  • In our example, the vertical intercept is 3, which means the line crosses the y-axis at the point (0, 3).
This point is the first step in graphing the line. From this point, you can use the slope to find other points on the line,
helping you draw the line accurately on a graph.
Horizontal Intercept
The horizontal intercept is another valuable concept when dealing with lines and their equations.
This is the point where the line crosses the x-axis.
  • To find the horizontal intercept, you set \( y = 0 \) in the equation of the line, because the x-axis is defined by \( y = 0 \).
  • Solving \( 0 = mx + b \) will give you the value of \( x \) where the line intercepts the x-axis.
In the provided example, the line equation is \( y = x + 3 \).
Setting \( y \) to zero, we have:

Finding the Horizontal Intercept

Set \( 0 = x + 3 \) and solve to find \( x = -3 \).Therefore, the horizontal intercept is at the point (-3, 0).
This intercept, combined with the vertical intercept, helps you fully understand how the line interacts with the axes.

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Most popular questions from this chapter

Budget constraints: Your family likes to eat fruit, but because of budget constraints, you spend only \(\$ 5\) each week on fruit. Your two choices are apples and grapes. Apples cost \(\$ 0.50\) per pound, and grapes cost \(\$ 1\) per pound. Let \(a\) denote the number of pounds of apples you buy and \(g\) the number of pounds of grapes. Because of your budget, it is possible to express \(g\) as a linear function of the variable \(a\). To find the linear formula, we need to find its slope and initial value. a. If you buy one more pound of apples, how much less money do you have available to spend on grapes? Then how many fewer pounds of grapes can you buy? b. Use your answer to part a to find the slope of \(g\) as a linear function of \(a\). (Hint: Remember that the slope is the change in the function that results from increasing the variable by 1. Should the slope of \(g\) be positive or negative?) c. To find the initial value of \(g\), determine how many pounds of grapes you can buy if you buy no apples. d. Use your answers to parts \(b\) and \(c\) to find \(a\) formula for \(g\) as a linear function of \(a\).

Competition between populations: In this exercise we consider the problem of competition between two populations that vie for resources but do not prey on each other. Let \(m\) be the size of the first population, let \(n\) be the size of the second (both measured in thousands of animals), and assume that the populations coexist eventually. An example of one common model for the interaction is Per capita growth rate for \(m\) is \(3(1-m-n)\) Per capita growth rate for \(n\) is $$ 2(1-0.7 m-1.1 n) $$ At an equilibrium point the per capita growth rates for \(m\) and for \(n\) are both zero. If the populations reach such a point, then they will continue at that size indefinitely. Find the equilibrium point in the example above.

Male and female high school graduates: The table below shows the percentage of male and female high school graduates who enrolled in college within 12 months of graduation. \({ }^{31}\) $$ \begin{array}{|l|c|c|c|c|} \hline \text { Year } & 1960 & 1965 & 1970 & 1975 \\ \hline \text { Males } & 54 \% & 57.3 \% & 55.2 \% & 52.6 \% \\ \hline \text { Females } & 37.9 \% & 45.3 \% & 48.5 \% & 49 \% \\ \hline \end{array} $$ a. Find the equation of the regression line for percentage of male high school graduates entering college as a function of time. b. Find the equation of the regression line for percentage of female high school graduates entering college as a function of time. c. Assume that the regression lines you found in part a and part b represent trends in the data. If the trends persisted, when would you expect first to have seen the same percentage of female and male graduates entering college? (You may be interested to know that this actually occurred for the first time in 1980 . The percentages fluctuated but remained very close during the \(1980 \mathrm{~s}\). In the 1990 s significantly more female graduates entered college than did males. In 1992 , for example, the rate for males was \(59.6 \%\) compared with \(63.8 \%\) for females.)

Speed of sound: The speed of sound in air changes with the temperature. When the temperature \(T\) is 32 degrees Fahrenheit, the speed \(S\) of sound is \(1087.5\) feet per second. For each degree increase in temperature, the speed of sound increases by \(1.1\) feet per second. a. Explain why speed \(S\) is a linear function of temperature \(T\). Identify the slope of the function. b. Use a formula to express \(S\) as a linear function of \(T\). c. Solve for \(T\) in the equation from part b to obtain a formula for temperature \(T\) as a linear function of speed \(S\). d. Explain in practical terms the meaning of the slope of the function you found in part \(c\).

An application of three equations in three unknowns: A bag of coins contains nickels, dimes, and quarters. There are a total of 21 coins in the bag, and the total amount of money in the bag is \(\$ 3.35\). There is one more dime than there are nickels. How many dimes, nickels, and quarters are in the bag?
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