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Chapter 3: Problem 78

(a) complete the table of solutions. (b) graph the equation. $$ \begin{aligned} &y=-3 x+1\\\ &\begin{array}{|r|r|} \hline x & y \\ \hline-2 & \\ \hline 0 & \\ \hline 2 & \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
The completed table is \[\begin{array}{|r|r|} \hline x & y \ \hline-2 & 7 \ \hline 0 & 1 \ \hline 2 & -5 \ \hline \end{array}\]. Plot the points \((-2, 7)\), \((0, 1)\), \(2, -5)\) and draw the line.

Step by step solution

01

Substitute \(x = -2\) in the equation

Use the equation \(y = -3x + 1\). Substituting \(x = -2\) gives \(y = -3(-2) + 1 = 6 + 1 = 7\). Therefore, when \(x = -2\), \(y = 7\).
02

Substitute \(x = 0\) in the equation

Using the equation \(y = -3x + 1\). Substitute \(x = 0\) to get \(y = -3(0) + 1 = 0 + 1 = 1\). Therefore, when \(x = 0\), \(y = 1\).
03

Substitute \(x = 2\) in the equation

Using the equation \(y = -3x + 1\). Substitute \(x = 2\) to get \(y = -3(2) + 1 = -6 + 1 = -5\). Therefore, when \(x = 2\), \(y = -5\).
04

Fill in the completed table

Now that the values of \(y\) have been calculated for each \(x\), the completed table is: \[\begin{array}{|r|r|} \hline x & y \ \hline-2 & 7 \ \hline 0 & 1 \ \hline 2 & -5 \ \hline \end{array}\]
05

Plot the points on the graph

The points to plot are \((-2, 7)\), \((0, 1)\), and \(2, -5)\). Mark these points on the coordinate plane.
06

Draw the line

Once the points are plotted, draw a straight line through these points. This line represents the equation \(y = -3x + 1\).

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Key Concepts

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

Table of Solutions
A table of solutions is used to find pairs of values for variables that satisfy a given equation. In this exercise, the equation is in the form of a linear equation: $$y = -3x + 1.$$ To complete the table, we substitute different values of $$x$$ into the equation and solve for $$y$$. Here鈥檚 how it鈥檚 done:
  • For $$x = -2$$, substitute $$x$$ into the equation: $$y = -3(-2) + 1 = 6 + 1 = 7.$$ So, the pair is \((-2, 7).\)
  • For $$x = 0$$, substitute $$x$$ into the equation: $$y = -3(0) + 1 = 0 + 1 = 1.$$ So, the pair is $$(0, 1).\(
  • For $$x = 2$$, substitute $$x$$ into the equation: $$y = -3(2) + 1 = -6 + 1 = -5.$$ So, the pair is $$(2, -5).\)
The completed table looks like this:
\begin{array}{|r|r|} \ hline x & y \ hline-2 & 7 \ hline 0 & 1 \ hline 2 & -5 \ hline \ end{array}
Each row shows an $$x$$ and $$y$$ pair that satisfy the equation.
Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane and drawing a line through these points. To graph the equation $$y = -3x + 1$$, we use the points obtained from the table of solutions: \((-2, 7)\), \((0, 1)\), and \((2, -5)\). Plot these points and connect them with a straight line. This visual representation helps to understand the relationship between $$x$$ and $$y$$. The line represents all the solutions to the equation.
Slope-Intercept Form
The slope-intercept form of a linear equation is written as $$y = mx + b$$, where:
  • $$m$$ is the slope of the line
  • $$b$$ is the y-intercept (the point where the line crosses the y-axis)
For the equation $$y = -3x + 1$$:
  • The slope ($$m$$) is $$-3$$. This means for every 1 unit increase in $$x$$, $$y$$ decreases by 3 units.
  • The y-intercept ($$b$$) is $$1$$. This is the value of $$y$$ when $$x$$ is 0.
Coordinate Plane
A coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). Each point on the plane is represented by an ordered pair $$(x, y)$$.
  • The horizontal distance from the origin (0,0) is given by $$x$$.
  • The vertical distance from the origin is given by $$y$$.
In this exercise, the points \((-2, 7)\), \((0, 1)\), and \((2, -5)\) are plotted on the coordinate plane to graph the line.
Plotting Points
Plotting points involves placing points on the coordinate plane according to their $$x$$ and $$y$$ values. Here鈥檚 how to plot the points:
  • Find \((-2, 7)\): Move 2 units to the left along the x-axis and 7 units up along the y-axis.
  • Find $$(0, 1)$$: Stay on the y-axis at 1 (no movement along the x-axis).
  • Find $$(2, -5)$$: Move 2 units to the right along the x-axis and 5 units down along the y-axis.
Connecting these points with a straight line gives the graph of the equation $$y = -3x + 1$$.

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Most popular questions from this chapter

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: Use the slope formula to find the slope of the line that passes through \((5,-9)\) and \((-2,-11)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{-11-9}{-2-5} \\ m &=\frac{-20}{-7} \\ m &=\frac{20}{7} \end{aligned} $$

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.

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((6,3)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{6}, 8\right) ;\left(\frac{5}{6}, 11\right)\)

(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. \((-1,-3) ;(-4,-1)\)
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