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Chapter 2: Problem 73

Graph each line passing through the given point and having the given slope. (0,0)\(; m=\frac{1}{5}\)

Short Answer

Expert verified
The equation of the line is \( y = \frac{1}{5}x \) and it passes through (0,0).

Step by step solution

01

- Understand the Slope-Intercept Form

Recall the slope-intercept form of a line, which is given by the equation: \[ y = mx + b \]where \(m\) is the slope and \(b\) is the y-intercept.
02

- Identify the Given Values

From the exercise, the given point is \( (0, 0) \)and the given slope is \( m = \frac{1}{5} \).
03

- Substitute the Point into the Equation

Since the point \((0,0)\) is given, it can be substituted into the slope-intercept form equation. We know that when \(x = 0\), \(y = 0\). Thus, substituting these into the equation \( y = \frac{1}{5}x + b \) gives \( 0 = \frac{1}{5}(0) + b \). Therefore, \( b = 0 \).
04

- Write the Equation of the Line

Now that the y-intercept \( b \) is found to be 0, substitute the values of \( m \) and \( b \) back into the slope-intercept form equation. This gives us the equation of the line: \[ y = \frac{1}{5}x \].
05

- Plot the Given Point

On a graph, plot the point \( (0, 0) \).
06

- Use the Slope to Determine Another Point

Using the slope \( m = \frac{1}{5} \), move 1 unit up and 5 units to the right from the given point \( (0, 0) \). This gives another point \( (5, 1) \).
07

- Draw the Line

Plot the second point \( (5, 1) \) on the graph. Draw a straight line through the points \( (0, 0) \) and \( (5, 1) \). This line represents the equation \( y = \frac{1}{5}x \).

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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 a way to express the equation of a straight line. It is given by: \[ y = mx + b \]where
  • m represents the slope of the line, showing how steep the line is
  • b is the y-intercept, the point where the line crosses the y-axis
In the exercise, the given slope is \( m = \frac{1}{5} \)and the point is \( (0, 0) \).Substituting \((0,0)\)into the formula yields \( b = 0 \). Thus, the equation of the line becomes \[ y = \frac{1}{5}x \].
Plotting Points
Plotting points on a graph involves placing dots at specific coordinates. For the exercise:
  • Start with the given point (0, 0): Using the coordinates, place a point at the origin of the graph.
  • Use the slope to find a second point: With a slope \( m = \frac{1}{5} \),one unit up and five units to the right leads to the point (5, 1).
Plot both points on the graph. This establishes the basic framework for the line.
Slope Calculation
The slope of a line indicates its steepness and direction. It's calculated by the formula: \[ m = \frac{rise}{run} \]where 'rise' is the change in the vertical direction and 'run' is the change in the horizontal direction.

For the given slope \( m = \frac{1}{5} \), it means:
  • Each step up (rise) by 1 unit
  • Corresponds to a 5-step move to the right (run)
Using these values, it's easy to identify new points from another known point, like (0, 0), giving the second point (5, 1).
When plotted, these points form a straight line that matches the slope \(\frac{1}{5}\).

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