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Chapter 3: Problem 26

Draw a linear graph to represent the given information. Be sure to label and number the axes appropriately. By 3: 00 PM., Arnie had earned \(\$ 70 .\) He continued earning money at a rate of \(\$ 12\) per hour.

Short Answer

Expert verified
Plot points (0, 70) and (1, 82) then draw a line. Use equation y = 12x + 70.

Step by step solution

01

Identify the Variables

First, identify the variables involved in the problem. Let the x-axis represent time (hours) and the y-axis represent Arnie's earnings (dollars).
02

Determine the Initial Point

At 3:00 PM, Arnie had earned \$70. This point will be represented as \((0, 70)\) on the graph, where x = 0 represents 3:00 PM.
03

Calculate the Slope

The rate at which Arnie earns money is \$$12/hour. This is the slope (m) of the line. Slope m = 12.
04

Write the Equation of the Line

Using the point-slope form of the equation \(y = mx + b\), where \m = 12\ and \b\ is the y-intercept (initial earnings), we get the equation:\( y = 12x + 70\).
05

Plot the Initial Point and Additional Points

Plot the initial point \(0, 70\). For additional points, use the equation y = 12x + 70. For example, after 1 hour (x=1), earnings are \y = 12(1) + 70 = \82. Plot the point \(1, 82\).
06

Draw the Line

Draw a straight line through the points plotted. This line represents Arnie's earnings over time.
07

Label and Number the Axes

Label the x-axis as 'Time (hours)' and y-axis as 'Earnings (dollars)'. Number the axes appropriately to accommodate the range of the values used in the graph.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form is vital in understanding linear equations. A linear equation can be expressed as \(y = mx + b\),
where \(m\) is the slope and \(b\) is the y-intercept. In this problem:
  • The slope \(m\) is \$12/hour, which tells us how much Arnie earns per hour.
  • The y-intercept \(b\) is \$70, representing his initial earnings at 3:00 PM.
Using the slope-intercept form, we can write the equation
\[y = 12x + 70\].
This equation helps in plotting points and drawing the linear graph. The slope indicates the steepness of the line, and the y-intercept tells us where the line starts on the y-axis.
Plotting Points
To create a graph, plotting points based on the equation is essential. First, identify the initial point. According to the problem, Arnie's earnings at 3:00 PM were \$70. This is plotted as the point \(0, 70\).
Now, use the equation \[y = 12x + 70\] to find more points. For instance:
  • After 1 hour (x=1), earnings are \[y = 12(1) + 70 = 82\]\(This gives the point \(1, 82\)\).
  • After 2 hours (x=2), earnings are \[y = 12(2) + 70 = 94\]\(This gives the point \(2, 94\)\).
By plotting these points on graph paper or a digital tool, you provide a visual representation of how Arnie's earnings increase over time.
Linear Graph
A linear graph helps visualize relationships between variables. In this case, it shows how time affects earnings. Start by labeling axes:
  • Label the x-axis as 'Time (hours)'.
  • Label the y-axis as 'Earnings (dollars)'.

Next, number the axes according to the points calculated (e.g., hours on the x-axis, dollars on the y-axis).
Then, plot your points like \(0, 70\), \(1, 82\), and \(2, 94\). Once the points are plotted, draw a straight line through them. This line represents the equation \[y = 12x + 70\]. The graph visually shows Arnie's earnings rising consistently by \$12 per hour.
Understanding this can give clarity on how changes in one variable impact another, a key concept in algebra.

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