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Chapter 3: Problem 42

(a) rewrite the equation in slope-intercept form. (b) identify the slope. (c) identify the \(y\)-intercept. Write the ordered pair, not just the \(y\)-coordinate. (d) find the \(x\)-intercept. Write the ordered pair, not just the \(x\)-coordinate. $$ 5 x-8 y=64 $$

Short Answer

Expert verified
Slope is \(\frac{5}{8}\), y-intercept is \((0, -8)\), x-intercept is \((12.8, 0)\).

Step by step solution

01

- Rewrite the equation in slope-intercept form

Start by getting the equation into the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Begin with the given equation:\[5x - 8y = 64\]Isolate \(y\). Subtract \(5x\) from both sides:\[-8y = -5x + 64\]Then divide all terms by \(-8\):\[y = \frac{5}{8}x - 8\]
02

- Identify the slope

The rewritten equation from Step 1 is:\[y = \frac{5}{8}x - 8\]Identify the slope \(m\), which is the coefficient of \(x\):\[m = \frac{5}{8}\]
03

- Identify the y-intercept

In the equation \(y = \frac{5}{8}x - 8\), the constant term is the \(y\)-intercept. Therefore, the y-intercept \(b = -8\). The ordered pair for the y-intercept is:\((0, -8)\)
04

- Find the x-intercept

To find the \(x\)-intercept, set \(y = 0\) and solve for \(x\) in the equation \(5x - 8y = 64\). Start from the original equation:\[5x - 8(0) = 64\]Simplify to find \(x\):\[5x = 64\]Divide both sides by 5:\[x = 12.8\]The ordered pair for the x-intercept is:\((12.8, 0)\)

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Key Concepts

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

Linear Equations
Linear equations are mathematical statements that show a direct relationship between two variables. These equations are represented in the form of a line on a graph. In general, linear equations take the form:\[ax + by = c\]where\(a\),\(b\), and\(c\) are constants.
Linear equations are classified as such because they can be graphically represented by a straight line.
These equations are foundational in algebra and can be used to solve real-world problems.
For example, the linear equation given in the exercise is:\[5x - 8y = 64\].
Slope
The slope of a line measures its steepness and direction. It's represented by the letter\(m\) in the slope-intercept form equation:\[y = mx + b\].
The slope is the ratio of the vertical change to the horizontal change between two points on the line.
Mathematically, slope is defined as:\[m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\].
In the given equation after converting it to slope-intercept form, \(y = \frac{5}{8}x - 8\), the slope\(m\) is identified as\(\frac{5}{8}\), signifying that for every 8 units moved horizontally to the right, the line moves 5 units vertically upward.
Intercepts
Intercepts are points where the line crosses the axes. There are two types of intercepts: y-intercept and x-intercept:
  • Y-intercept: The point where the line crosses the y-axis, represented as \((0, b)\) in slope-intercept form. In our example, \(y = \frac{5}{8}x - 8\), the y-intercept \(b\) is -8, giving the ordered pair \((0, -8)\).

  • X-intercept: The point where the line crosses the x-axis. To find it, set \(y = 0\) and solve for \(x\). From our equation, setting \(y = 0\) in \(5x - 8y = 64\) and solving gives the x-intercept as \((12.8, 0)\).
Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of linear equations, algebra helps us transform equations and solve for unknown variables. Using algebra, we converted the given equation:\[5x - 8y = 64\],into its slope-intercept form to make it easier to find the slope and intercepts. Here鈥檚 how step-by-step algebraic manipulation works:
  • Start with \(5x - 8y = 64\)
  • Isolate \(y\) by subtracting \(5x\) from both sides: \(-8y = -5x + 64\)
  • Divide every term by \(-8\): \(y = \frac{5}{8}x - 8\)
These steps improve our understanding of the relationship between variables in the equation.

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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.

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

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(y=-x+3\)

Use a graphing calculator to graph each equation. Choose a window that shows the \(x\)-intercept and \(y\)-intercept. Sketch the graph; describe the window. \(y=2 x+5\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{9}{7}\right) ;\left(\frac{5}{3}, \frac{3}{7}\right)\)
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