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Chapter 3: Problem 61

(a) complete the table of solutions. (b) graph the equation. $$ \begin{aligned} &x-6 y=9\\\ &\begin{array}{|l|l|} \hline x & y \\ \hline 0 & \\ \hline & 0 \\ \hline 3 & \\ \hline \end{array} \end{aligned} $$

Short Answer

Expert verified
The table of solutions is \(\begin{array}{|l|l|} \hline 0 & -\frac{3}{2} \ \hline 9 & 0 \ \hline 3 & -1 \ \hline \end{array}\). The graph is a line through these points.

Step by step solution

01

- Solve for y when x = 0

Substitute x = 0 into the equation: \(0 - 6y = 9\) Divide both sides by -6: \(y = -\frac{3}{2}\)
02

- Solve for x when y = 0

Substitute y = 0 into the equation: \(x - 6(0) = 9\) Simplify: \(x = 9\)
03

- Solve for y when x = 3

Substitute x = 3 into the equation: \(3 - 6y = 9\) Subtract 3 from both sides: \(-6y = 6\) Divide both sides by -6: \(y = -1\)
04

- Complete the table

Based on the computations, the completed table is: \(\begin{array}{|l|l|} \hline x & y \ \hline 0 & -\frac{3}{2} \ \hline 9 & 0 \ \hline 3 & -1 \ \hline \end{array}\)
05

- Graph the equation

To graph the equation, plot the points (0, -1.5), (9, 0), and (3, -1). Draw a straight line that passes through these points as the graph of the equation.

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Key Concepts

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

Solving Linear Equations
Let's dive into solving linear equations. A linear equation is an equation involving variables like x or y, where the highest power of the variable is one. For a simple example, take the equation: \(x - 6y = 9\). Solving this equation means finding the values of x and y that make the equation true.

Here's the step-by-step approach we used for solving:
  • Identify which variable you want to solve for first.
  • Substitute a value, usually zero, for the other variable to simplify calculations.
  • Solve the simplified equation to get the value of the chosen variable.
  • Repeat the steps for the other variable.
Clear and simple, right? Once you find these values, you can check your work by substituting them back into the original equation to see if it holds true.
Understanding linear equations is fundamental to solving and graphing them properly.
Graphing Linear Equations
Graphing linear equations helps you see their solutions visually. The equation \(x - 6y = 9\) can be represented on a graph by plotting specific points that satisfy the equation. These points lie on a straight line.

Here鈥檚 how we graph it step by step:
  • First, we completed the table of values by solving for y when x = 0, x when y = 0, and y when x = 3.
  • Then, we used the resulting points (0, -1.5), (9, 0), and (3, -1) to graph the line.
  • Plot each point on the coordinate plane accurately.
  • Draw a straight line through the plotted points. This line represents our linear equation.
By graphing, you turn algebraic equations into visual lines that make it easier to understand relationships between variables.
Substitution Method
The substitution method is a key technique for solving linear equations. You use it to find the value of one variable by substituting a known value of another variable. For the equation \(x - 6y = 9\), we did the following:

To solve for y when x = 0:
  • Substitute x = 0 into the equation: \(0 - 6y = 9\).
  • Simplify to get \(-6y = 9\).
  • Finally, solve for y by dividing both sides by -6, thus \(y = -\frac{3}{2}\).
This method involves isolating one variable and then making the substitution to solve for the other. It鈥檚 straightforward and hugely useful in algebra. You can swap steps for different variables based on what鈥檚 convenient for the specific problem you are solving.
Coordinate Plotting
Coordinate plotting is a fundamental skill in graphing linear equations. To plot the coordinates, you follow these steps:

  • First, identify the coordinates you want to plot from your solved values.
  • Plot the first coordinate, such as (0, -1.5), on the graph by moving 0 units along the x-axis and -1.5 units along the y-axis.
  • Repeat for other points, for example, (9, 0) means you move 9 units along the x-axis and 0 units along the y-axis.
  • Continue plotting all identified points accurately.
Use a ruler to draw a line through these points.
Coordinating plotting transforms algebraic solutions into visible lines on a graph, making it easier to understand the relationship between x and y. Remember, accuracy is key for a correct representation!

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Most popular questions from this chapter

For exercises 9鈥�20, (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. \((1,4) ;(3,10)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(-8, \frac{1}{4}\right) ;\left(16, \frac{1}{2}\right)\)

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) 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. \((1,4) ;(3,-6)\)

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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