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Chapter 3: Problem 71

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

Short Answer

Expert verified
Three solutions are (1,1), (2,2), and (-1,-1). Graph the line y = x.

Step by step solution

01

- Understand the Equation

The equation given is a linear equation of the form y = x. This means for every value of x, y will have the same value.
02

- Find Three Solutions

To find three solutions, simply choose three different values for x and compute the corresponding y values. For example:1. When x = 1, y = 1.2. When x = 2, y = 2.3. When x = -1, y = -1.
03

- Plot the Points

Using the three solutions found in the previous step, plot these points on a graph. The points are (1,1), (2,2), and (-1,-1).
04

- Draw the Line

Draw a straight line through the plotted points. This line represents the equation y = x.
05

- Label the Graph

Label the x-axis and y-axis appropriately, and ensure the line is extended in both positive and negative directions, showing that the relationship y = x holds for all real numbers.

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Key Concepts

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

graphing equations
Graphing equations is a powerful way to visualize mathematical relationships. To graph an equation, you first need to understand the relationship between the variables involved. In the case of the linear equation y = x, the relationship is straightforward: for every value of x, y has the same value. This equation forms a straight line on a graph. To start:
  • Select a few values for x.
  • Compute the corresponding y values.
  • Plot the points on a coordinate plane and connect them to reveal the line.
Notice how the plotted points form a straight line, showing the consistent relationship between x and y.
solutions of equations
A solution to an equation is a set of values for variables that satisfy the equation. For linear equations, solutions are pairs of values (x, y) that make the equation true. Consider the equation y = x. To find solutions:
  • Choose different x values. These are often integers for simplicity.
  • Calculate the corresponding y values using the equation.
For example, if you select x = 1, 2, and -1, you find the solutions (1, 1), (2, 2), and (-1, -1) respectively. These points show specific solutions where both sides of the equation balance.
plotting points
Plotting points is an essential skill in graphing equations. Each point represents a solution and is written as (x, y). To plot points:
  • Start at the origin (0,0) where the x-axis and y-axis intersect.
  • Move horizontally to the x value.
  • Then move vertically to the y value.
Let's plot the points (1, 1), (2, 2), and (-1, -1) given their solutions:
  • From the origin, move to x=1 and then up to y=1 for the point (1, 1).
  • Next, move to x=2 and then up to y=2 for the point (2, 2).
  • For (鈭�1,鈭�1), move left to x=鈭�1, then down to y=鈭�1.
These points help represent the line y = x visually.
understanding linear relationships
Linear relationships describe a direct connection between two variables. In the equation y = x, the relationship is simple: y increases as x increases, equally. This linear relationship can be identified because:
  • The equation can be written in the form y = mx + b where m is the slope and b is the y-intercept.
  • Here, m = 1 and b = 0, meaning the line passes through the origin at a 45-degree angle.
  • The slope (m) of 1 indicates a one-to-one increase in both variables.
By plotting several points, the pattern becomes clear, reflecting the steady, equal increments of both x and y.

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Most popular questions from this chapter

A large garden is a collection of 24 raised beds. Each bed is a rectangle that is \(8 \mathrm{ft}\) long and \(4 \mathrm{ft}\) wide. Find the total area of the raised beds.

(a) graph the given points, and draw a line through the points. (b) use the graph to find the slope of the line. (c) use the slope formula to find the slope of the line. \((-2,-5) ;(1,3)\)

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.

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: \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\) $$ \begin{aligned} m &=\frac{9-7}{4-1} \\ m &=\frac{2}{3} \end{aligned} $$

For exercises 37-66, use the slope formula to find the slope of the line that passes through the points. \((1,9) ;(3,14)\)
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