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Chapter 3: Problem 40

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(5 x-2 y=20\)

Short Answer

Expert verified
y-intercept: (0, -10), x-intercept: (4, 0), another solution: (2, -5).

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set x = 0 in the equation 5x - 2y = 20 and solve for y. 5(0) - 2y = 20 -2y = 20 y = -10 . So, the y-intercept is (0, -10).
02

- Find the x-intercept

To find the x-intercept, set y = 0 in the equation 5x - 2y = 20 and solve for x. 5x - 2(0) = 20 5x = 20 x = 4 . So, the x-intercept is (4, 0).
03

- Find a third solution

We can choose any value for x or y and solve for the other. Let's choose x = 2. Substitute x = 2 into the equation 5x - 2y = 20 and solve for y. 5(2) - 2y = 20 10 - 2y = 20 -2y = 10 y = -5 . So, another solution is (2, -5).
04

- Graph the equation

Plot the points found: (0, -10), (4, 0) and (2, -5) on a graph. Draw a straight line through these points, extending the line in both directions. This represents the graph of the equation 5x - 2y = 20.

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Key Concepts

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

y-intercept
The y-intercept is where the line crosses the y-axis. To find it, set x to 0 in the equation and solve for y. For the equation \(5x - 2y = 20\), substituting x = 0 gives us:\[5(0) - 2y = 20\]\[-2y = 20\]\[y = -10\]This means the y-intercept is (0, -10).
x-intercept
The x-intercept is where the line crosses the x-axis. To find it, set y to 0 in the equation and solve for x. For the equation \(5x - 2y = 20\), substituting y = 0 gives us:\[5x - 2(0) = 20\]\[5x = 20\]\[x = 4\]This means the x-intercept is (4, 0).
solving linear equations
Solving linear equations often involves finding particular points, such as intercepts. For instance, if you randomly choose a value for x, say x = 2, you can solve for y in the equation \(5x - 2y = 20\):\[5(2) - 2y = 20\]\[10 - 2y = 20\]\[-2y = 10\]\[y = -5\].These calculations show another solution to the equation: (2, -5).
graphing coordinates
Graphing a linear equation involves plotting points like the intercepts and any additional solutions, then drawing a straight line through these points. Using the points found (0, -10), (4, 0), and (2, -5), plot them on a graph. Draw a line that fits all these points, extending it in both directions. This shows the graphical representation of \(5x - 2y = 20\).

Remember:
  • The y-intercept is (0, -10)
  • The x-intercept is (4, 0)
  • Another point is (2, -5)

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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: Find the \(x\)-intercept of \(9 x+2 y=36\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

Use the slope formula to find the slope of the line that passes through the points. \((-1,5) ;(-6,-13)\)

For exercises 95-98, 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 \(y\)-intercept of \(6 x+5 y=30\). Incorrect Answer: \(9 x+2 y=36\) $$ \begin{aligned} 9(0)+2 y &=36 \\ 2 y &=36 \\ \frac{2 y}{2} &=\frac{36}{2} \\ y &=18 \end{aligned} $$ The \(x\)-intercept is \((0,18)\).

Use the slope formula to find the slope of the line that passes through the points. \((-5,-4) ;(-9,-3)\)

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.
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