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Chapter 3: Problem 22

A line passes through the given points. (a) Find the slope of the line. (b) Write the equation of the line in slope-intercept form. $$ (5,9) ;(7,17) $$

Short Answer

Expert verified
(a) The slope is 4. (b) The equation is \ y = 4x - 11 \.

Step by step solution

01

- Identify the given points

The given points are \( (5, 9) \) and \( (7, 17) \). Represent the first point as \( (x_1, y_1) = (5, 9) \) and the second point as \( (x_2, y_2) = (7, 17) \).
02

- Find the slope of the line

Use the formula for the slope \( m \) between two points: \ m = \frac{y_2 - y_1}{x_2 - x_1} \. Substituting the coordinates of the points, the slope is \ m = \frac{17 - 9}{7 - 5} = \frac{8}{2} = 4 \.
03

- Use the slope-intercept form

The slope-intercept form of a line is given by \ y = mx + b \. Use one of the points \( (5, 9) \) and the slope \( m = 4 \) to find the y-intercept \( b \).
04

- Substitute the values into the slope-intercept form

Substitute \( x = 5 \) and \( y = 9 \) into the equation \( y = 4x + b \): \ 9 = 4(5) + b \. Simplifying, \ 9 = 20 + b \ leads to \ b = 9 - 20 = -11 \.
05

- Write the equation of the line

Insert the slope \( m = 4 \) and the y-intercept \( b = -11 \) into the slope-intercept form: \ y = 4x - 11 \.

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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 way of writing the equation of a straight line. This form is very useful because it immediately gives the slope and the y-intercept of the line. The slope-intercept form of a line is written as \(y = mx + b\), where:
  • \(m\) is the slope of the line
  • \(b\) is the y-intercept
The slope indicates how steep the line is. The y-intercept is the point where the line crosses the y-axis. In the equation \(y = 4x - 11\), the slope \(m\) is 4, and the y-intercept \(b\) is -11. This means the line rises 4 units for every 1 unit it moves to the right and it crosses the y-axis at -11.
finding the slope
The slope of a line is a measure of how steep it is. It is calculated as the change in the y-values divided by the change in the x-values between two points on the line. The formula for finding the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's use our points \((5, 9)\) and \((7, 17)\). Substituting these into the formula:
\[ m = \frac{17 - 9}{7 - 5} = \frac{8}{2} = 4 \] So, the slope of the line is 4. This means the line rises 4 units for every 1 unit it moves to the right.
linear equations
Linear equations represent straight lines on a graph. These equations can be written in various forms, but the most commonly used is the slope-intercept form, \(y = mx + b\). To find the equation of a line from two points, we:
  • Calculate the slope \(m\)
  • Use one of the points to solve for the y-intercept \(b\)
Here's how we did it for the points \((5, 9)\) and \((7, 17)\):
  • First, we found the slope to be 4.
  • Next, we used the slope and one of the points to find the y-intercept:
From the point \((5, 9)\), we substitute into the slope-intercept form \(y = mx + b\):
\[ 9 = 4(5) + b \rightarrow 9 = 20 + b \rightarrow b = 9 - 20 \rightarrow b = -11 \] So, the equation of the line is \(y = 4x - 11\). This allows anyone to plot the line on a graph or see how the line behaves for different values of \(x\).

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Most popular questions from this chapter

Some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Have you noticed anything that your instructor does while teaching that you find helps you remember what has been taught?

For exercises 97-98, some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any of the strategies listed above that you currently use to study math.

Explain why the \(x\)-intercept, the \(y\)-intercept, and one other point are not three different solutions of \(y=2 x\).

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-y=7\)

For exercises 89-92, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Find the slope of the line that passes through \((7,1)\) and \((9,4)\). Incorrect Answer: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$
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