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Chapter 3: Problem 67

(a) find three solutions of the equation. (b) graph the equation. \(y=-x+4\)

Short Answer

Expert verified
Three solutions are (0, 4), (2, 2), and (-1, 5). To graph the equation, plot these points and draw a line through them.

Step by step solution

01

Identify the equation

The given equation is a linear equation in the form: \(y = -x + 4\).
02

- Find three solutions

To find three solutions, substitute three different values for \(x\) into the equation and solve for \(y\): 1. Let \(x = 0\), then \(y = -0 + 4 = 4\), so the solution is \((0, 4)\). 2. Let \(x = 2\), then \(y = -2 + 4 = 2\), so the solution is \((2, 2)\). 3. Let \(x = -1\), then \(y = -(-1) + 4 = 5\), so the solution is \((-1, 5)\).
03

- Prepare to graph

Write down the points found in Step 1: \((0, 4)\), \((2, 2)\), and \((-1, 5)\).
04

- Plot the points

On graph paper or a coordinate plane, plot the points \((0, 4)\), \((2, 2)\), and \((-1, 5)\).
05

- Draw the line

Connect the points with a straight line. This line represents the equation \(y = -x + 4\).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in algebra. In our example, the given equation is in slope-intercept form: \(y = -x + 4\). This form, \(y = mx + b\), tells us two things: the slope \(m\) and the y-intercept \(b\). Here, the slope \(m\) is \(-1\) and the y-intercept \(b\) is \(4\).

Follow these steps to graph the equation:
  • Step 1: Start by plotting the y-intercept. Locate the point where the line crosses the y-axis (\(y = 4\)).
  • Step 2: Use the slope to find more points. The slope of \(-1\) means that for every unit you move to the right, you move one unit down. So from (0,4), go right 1 and down 1 to get to (1,3).
  • Step 3: Plot additional points if required.
  • Step 4: Draw a straight line connecting these points. This line is the graphical representation of the given linear equation.
By consistently following these steps, you'll be able to graph any linear equation without difficulties.
Finding Solutions to Equations
Finding solutions to a linear equation involves identifying pairs of \((x, y)\) that satisfy the equation. In the equation \(y = -x + 4\), we can substitute different values of \(x\) to find corresponding values of \(y\).

Let's go step-by-step:
  • Choose any value for \(x\): This can be anything, but choosing simple numbers can make calculations easier.
  • Substitute \(x\) into the equation: For instance, if \(x = 0\), then \(y = -0 + 4 = 4\), producing the solution \((0, 4)\).
  • Calculate the corresponding \(y\) value: If \(x = 2\), then \(y = -2 + 4 = 2\), giving us the solution \((2, 2)\).
  • Repeat: Try another value like \(x = -1\), leading to \(y = -(-1) + 4 = 5\), giving us \((-1, 5)\).
By choosing different \(x\) values and substituting them into the equation, we find multiple \((x, y)\) pairs, known as 'points.' At least three points help us draw an accurate graph.
Coordinate Plane
A coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical), intersecting at the origin (0, 0).

Here's how you can use a coordinate plane to plot points and graph an equation:
  • Understand the axes: The horizontal line is called the x-axis, and the vertical line is the y-axis.
  • Label the axes and origin: Mark the point (0,0) where the two axes intersect.
  • Plot points: Using the solutions we found, place each point on the plane. For example, plot (0, 4) by starting at origin, moving zero units horizontally, and four units upward.
  • Check the placement: Ensure each point is accurately placed reflecting both the x and y values.
  • Connect the points: Draw a straight line through all plotted points to represent the linear equation graphically.
Mastering the use of a coordinate plane is vital for visualizing relationships between variables and understanding the graphical aspect of linear equations.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(x-4 y=48\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{8}, 7\right) ;\left(\frac{7}{8}, 9\right)\)

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.

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 strategies listed that you don't currently use but you think might be helpful.

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The estimated number of wireless connections in the United States increased from 207,896,198 connections in 2005 to \(302,859,674\) connections in 2010 . Round to the nearest thousand. (Source: www.ctia.org)
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