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Chapter 3: Problem 71

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(-5 x+2 y=40\)

Short Answer

Expert verified
The y-intercept is (0, 20), the x-intercept is (-8, 0), and the slope is \(\frac{5}{2}\).

Step by step solution

01

Finding the y-intercept

To find the y-intercept, set the value of x to 0 in the equation \(-5x + 2y = 40\). Substitute x=0 into the equation: \(-5(0) + 2y = 40\). Simplifying it, we get: \(2y = 40\). Divide both sides by 2 to solve for y: \(y = 20\). Thus, the y-intercept is (0, 20).
02

Finding the x-intercept

To find the x-intercept, set the value of y to 0 in the equation \(-5x + 2y = 40\). Substitute y=0 into the equation: \(-5x + 2(0) = 40\). Simplifying it, we get: \(-5x = 40\). Divide both sides by -5 to solve for x: \(x = -8\). Thus, the x-intercept is (-8, 0).
03

Finding the slope

To find the slope, first put the equation into slope-intercept form (\(y = mx + b\)). Start with the given equation: \(-5x + 2y = 40\). Isolate y by adding \(5x\) to both sides: \(2y = 5x + 40\). Then divide by 2: \(y = \frac{5}{2}x + 20\). So, the slope \(m\) is \(\frac{5}{2}\).

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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 on a graph. To find this, we set the value of x to 0 in the given equation. For the equation \(-5x + 2y = 40\), setting x to 0 gives us \(-5(0) + 2y = 40\). This simplifies to \(2y = 40\). By dividing both sides by 2, we get \(y = 20\). Therefore, the y-intercept is the point \((0, 20)\). This means the line crosses the y-axis at the coordinate (0, 20).
x-intercept
The x-intercept is where the line crosses the x-axis on a graph. To find this, we set the value of y to 0 in the given equation. For the equation \(-5x + 2y = 40\), setting y to 0 gives us \(-5x + 2(0) = 40\). This simplifies to \(-5x = 40\). By dividing both sides by -5, we get \(x = -8\). Therefore, the x-intercept is the point \((-8, 0)\). This means the line crosses the x-axis at the coordinate (-8, 0).
slope formula
The slope of a line measures its steepness and is usually represented by 'm'. The slope formula is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). However, another common way to find the slope is to convert the equation into slope-intercept form. For the equation \(-5x + 2y = 40\), we first isolate y. Adding \(5x\) to both sides gives us \(2y = 5x + 40\). We then divide both sides by 2 to get \(y = \frac{5}{2}x + 20\). Here, \(\frac{5}{2}\) is the slope, so the slope of the line is \(\frac{5}{2}\).
slope-intercept form
The slope-intercept form of a linear equation is given by \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept. For the equation \-5x + 2y = 40\, we convert it to slope-intercept form by isolating y. Adding \(5x\) to both sides gives us \(2y = 5x + 40\). Dividing both sides by 2, we get \(y = \(\frac{5}{2}\)x + 20\). Thus, the equation in slope-intercept form is \(y = \(\frac{5}{2}\)x + 20\), where \(\frac{5}{2}\) is the slope (m) and 20 is the y-intercept (b). This form makes it easy to graph the line and understand its features.

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Most popular questions from this chapter

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.

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 \((6,2)\) and \((6,7)\). $$ \text { Incorrect Answer: } \begin{aligned} m &=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m &=\frac{6-6}{7-2} \\ m &=\frac{0}{5} \\ m &=0 \end{aligned} $$

Balanced Rock in Arches National Park is \(55 \mathrm{ft}\) tall and weighs 3500 tons. Find its height in meters. Round to the nearest tenth. \((1 \mathrm{~m} \approx 3.2808 \mathrm{ft}\).) (Source: www.desertusa .com)

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. \((-2,-5) ;(1,3)\)
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