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Chapter 3: Problem 58

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(30 x+y=600\)

Short Answer

Expert verified
y-intercept: (0, 600). x-intercept: (20, 0). Another solution: (10, 300).

Step by step solution

01

Find the y-intercept

To find the y-intercept, set x = 0 and solve for y. Substitute x = 0 in the equation: \[ 30(0) + y = 600 \] This simplifies to: \[ y = 600 \] So, the y-intercept is (0, 600).
02

Find the x-intercept

To find the x-intercept, set y = 0 and solve for x. Substitute y = 0 in the equation: \[ 30x + 0 = 600 \] This simplifies to: \[ 30x = 600 \] Divide both sides by 30: \[ x = 20 \] So, the x-intercept is (20, 0).
03

Find a third solution

Choose a value for x or y to find a third point on the line. Let's choose x = 10 and solve for y. Substitute x = 10 in the equation: \[ 30(10) + y = 600 \] This simplifies to: \[ 300 + y = 600 \] Subtract 300 from both sides: \[ y = 300 \] So, another solution is (10, 300).
04

Graph the equation

Plot the intercepts and the third solution on a coordinate plane. Plot the points (0, 600), (20, 0), and (10, 300). Draw a straight line through these points to represent the equation \[ 30x + y = 600 \].

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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 of a linear equation is the point where the line crosses the y-axis. This happens when the value of x is zero. To find the y-intercept in the equation 30x + y = 600, we set x to 0 and solve for y.
This simplifies to:
\[ 30(0) + y = 600 \]
which becomes:
\[ y = 600 \]
Therefore, the y-intercept is at the point (0, 600). This means the line crosses the y-axis at 600.

Recognizing the y-intercept is useful because it gives a starting point to begin graphing the line.
x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. This happens when the value of y is zero. To find the x-intercept in the equation 30x + y = 600, we set y to 0 and solve for x.
This simplifies to:
\[ 30x + 0 = 600 \]
which becomes:
\[ 30x = 600 \]
Divide both sides by 30:
\[ x = 20 \]
Therefore, the x-intercept is at the point (20, 0).
This means the line crosses the x-axis at 20.

Knowing the x-intercept is helpful when graphing because it provides another key point on the line.
graphing linear equations
Graphing a linear equation involves plotting points that satisfy the equation and drawing a line through them.
Here are the steps for graphing:
  • Find the intercepts: For the equation 30x + y = 600, we've already determined the y-intercept (0, 600) and the x-intercept (20, 0).
  • Find a third point: Choose a value for x or y to find another point on the line. We chose x = 10 earlier and found y = 300, giving us the point (10, 300).
Plot these points on a coordinate plane:
  • (0, 600)
  • (20, 0)
  • (10, 300)
Draw a straight line through these points to represent the equation \[ 30x + y = 600 \].
This line gives a visual representation of all solutions to the equation.
finding solutions
Finding solutions to a linear equation means locating points (x, y) that satisfy the equation. Each solution represents a point on the line.
To find these solutions, you can:
  • Use the intercepts: The x-intercept and y-intercept are easy solutions.
  • Choose a value for x or y: Use substitution to find the corresponding value. For example, choosing x = 10 and solving for y in 30x + y = 600 gave us y = 300, producing the point (10, 300).

Each solution provides a pair of coordinates indicating a point on the line.

By selecting different values of x or y, you can find multiple points that all lie on the same line, confirming the linear relationship described by the equation.

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Most popular questions from this chapter

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.

Use the slope formula to find the slope of the line that passes through the points. \((3,11) ;(7,39)\)

Use the slope formula to find the slope of the line that passes through the points. \((-3,8) ;(11,23)\)

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

Use the slope formula to find the slope of the line that passes through the points. \((4 \mathrm{hr}, \$ 36) ;(6 \mathrm{hr}, \$ 54)\)
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