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Chapter 1: Problem 23

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(-2,\) slope \(-\frac{1}{2}\)

Short Answer

Expert verified
\( y = -\frac{1}{2}x -1 \)

Step by step solution

01

Identify the required formula

The equation of a line in slope-intercept form is given by: \[ y = mx + b \] where \( m \) is the slope and \( b \) is the \( y \)-intercept.
02

Understand the given information

The problem provides the \( x \)-intercept and the slope. The \( x \)-intercept is \( -2 \) and the slope \( m \) is \( -\frac{1}{2} \).
03

Use the \( x \)-intercept to find the \( y \)-intercept

At the \( x \)-intercept, the value of \( y \) is 0. Thus, we start with the given \[ y = mx + b \] Replace \( y \) with 0 and \( x \) with \( -2 \):\[ 0 = -\frac{1}{2}(-2) + b \]
04

Solve for the \( y \)-intercept

Solving the equation from step 3:\[ 0 = 1 + b \] Subtract 1 from both sides:\[ b = -1 \]
05

Write the final equation of the line

Using the values of \( m \) and \( b \), substitute them back into the slope-intercept form:\[ y = -\frac{1}{2}x -1 \]

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Key Concepts

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

x-intercept
The x-intercept of a line is where it crosses the x-axis. At this point, the value of y is 0. To find the x-intercept, you set y to 0 in the equation of the line and solve for x. It is very useful when trying to understand the behavior of a line in a 2D coordinate system. For example, if a line crosses the x-axis at -2, it means that at the point (-2, 0), the line touches the axis. Remember, the x-intercept provides a specific point that helps in graphing and understanding lines.
slope-intercept form
The slope-intercept form is one of the most common ways to write the equation of a line. It is represented as \( y = mx + b \). Here, 'm' stands for the slope of the line, and 'b' is the y-intercept. This form is very straightforward and makes it easy to graph the line since you can start at the y-intercept and use the slope to find other points. Knowing the slope-intercept form is essential for quickly solving and graphing linear equations.
y-intercept
The y-intercept of a line is when it crosses the y-axis. At this point, x is 0. Finding the y-intercept is often simple if you have the equation in slope-intercept form \( y = mx + b \). Just look at the value 'b'. For example, in the equation \( y = -\frac{1}{2}x -1 \), the y-intercept is -1. At (0, -1), the line crosses the y-axis. Understanding the y-intercept helps you start graphing a line properly.
algebra
Algebra is the branch of mathematics dealing with symbols and rules for manipulating those symbols. It's the foundation for understanding linear equations. Learning algebra allows you to solve for unknown values, manipulate equations, and understand mathematical relationships. When dealing with linear equations, you'll perform operations like addition, subtraction, multiplication, and division to isolate variables and find their values.
equation of a line
The equation of a line tells you everything you need to know about that line. In the slope-intercept form \( y = mx + b \), 'm' represents the slope, indicating how steep the line is and which direction it goes. 'b' is the y-intercept, the point where the line crosses the y-axis. Knowing how to derive and interpret the equation allows you to graph the line, understand its characteristics, and solve various problems related to it. For example, given the x-intercept -2 and slope -1/2, you can find the y-intercept by setting y to 0 and solving for b in the equation \( y = mx + b \).

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Most popular questions from this chapter

In this exercise, you will apply what you have learned about writing equations for parallel lines. a. Write three equations whose graphs are parallel lines with positive slopes. Write the equations so that the graphs are equally spaced. b. Graph the lines, and verify that they are parallel. c. Write three equations whose graphs are parallel lines with negative slopes and are equally spaced. d. Graph the lines, and verify that they are parallel.

The Glitz mail order company charges \(\$ 1.75\) per pound for shipping and handling on customer orders. The Lusterless mail order company charges \(\$ 1.50\) per pound for shipping and handling, plus a flat fee of \(\$ 1.25\) for all orders. a. For each company, make a table showing the costs of shipping items of different whole-number weights from 1 to 10 pounds. b. Write an equation for each company to help calculate how much you would pay for shipping, C, on an order of any weight, W. c. Draw graphs of your equations, and label each with the corre- sponding company鈥檚 name. d. Which company offers the better deal on shipping? e. Describe how the graphs you drew could help you answer Part d. f. How would the Lusterless company have to change their rates to make them vary directly with the weight of a customer鈥檚 order?

Consider this data set. $$\begin{array}{|c|c|c|c|c|c|}\hline x & {0} & {2} & {4} & {6} & {8} \\\ \hline y & {2} & {20} & {6} & {8} & {10} \\ \hline\end{array}$$ a. Graph the data set. b. One point is an outlier. Which point is it? c. Find the mean of the x values and the mean of the y values. d. Try to find a line that is a good fit for the data and goes through the point (mean of \(x\) values, mean of \(y\) values). Write an equation for your line. e. Now find the means of the variables, ignoring the outlier. In other words, do not include the values for the outlier in your calculations. f. Try to find a new line that is a good fit for the data, using the means you calculated in Part e for the (mean of x values, mean of y values) point. Write an equation for your line. g. Do you think either line should be considered the best fit for the data? Explain.

Alejandro looked at the equations \(y=\frac{3}{2} x-1\) and \(y=-\frac{2}{3} x+2\) and said, These lines form a right angle. a. Graph both lines on two different grids, with the axes labeled as shown here. b. Compare the lines on each grid. Do both pairs of lines form a right angle? c. What kind of assumption must Alejandro have made when he said the lines form a right angle?

Just as the \(y\) -intercept of a line is the \(y\) value at which the line crosses the \(y\) -axis, the \(x\) -intercept is the \(x\) value at which the line crosses the \(x\) -axis. In Exercises \(22-25,\) find an equation of a line with the given \(x\) -intercept and slope. \(x\) -intercept \(3,\) no slope (Hint: If slope is \(\frac{\text { rise }}{\text { run }}\) , when would there be no slope?)
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