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Chapter 7: Problem 82

Graph each line passing through the given point and having the given slope $$ (5,3) ; m=0 $$

Short Answer

Expert verified
The equation of the line is \( y = 3 \).

Step by step solution

01

Understand the Slope

The slope provided is 0. This means that the line is horizontal, and there is no change in the y-value regardless of the x-value.
02

Identify the Point

The line passes through the point (5, 3). This point is significant because it helps to locate the line on the graph.
03

Write the Equation of the Line

Since the slope (m) is 0, the equation of the line can be expressed as a horizontal line passing through the y-value of the given point. Therefore, the equation is: \[ y = 3 \]
04

Graph the Line

To graph the line, draw a horizontal line that crosses the y-axis at y = 3. This line runs parallel to the x-axis.

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Key Concepts

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

Slope of a Line
The slope of a line, often denoted as \(m\), is a measure of the steepness or incline of the line. It represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
If the slope is positive, the line ascends from left to right. If the slope is negative, the line descends from left to right.
A zero slope, like in this exercise, means the line is horizontal.
Key points about slope:
  • A positive slope rises as it moves to the right.
  • A negative slope falls as it moves to the right.
  • A zero slope indicates a horizontal line.
  • An undefined slope indicates a vertical line.
Equation of a Line
The equation of a line can generally be written in the format:
\( y = mx + b \),
where \(m\) is the slope and \(b\) is the y-intercept, the point where the line crosses the y-axis.
In this exercise, the slope \(m\) is 0, simplifying the equation to \( y = b \). Since we know the line passes through the point (5, 3), the y-intercept \(b\) is 3. Therefore, the equation of the line is: \[ y = 3 \] The equation tells us that no matter the value of \(x\), the \(y\)-value remains 3, confirming the line is horizontal.
Horizontal Lines
Horizontal lines are special because they have a slope of zero. They run parallel to the x-axis and do not rise or fall. When given a point and a zero slope, like in this exercise, the equation of the line is simple to find. It has the form:
\( y = c \),
where \(c\) is the constant y-value for all points on the line.
For example, the line passing through (5, 3) with a slope of 0 is \( y = 3 \). Right along the y-value 3, this line extends infinitely in both directions parallel to the x-axis.
To graph a horizontal line, just draw a straight line through the given y-value. Each point on this line has the same y-coordinate.
  • Example: \( y = 2 \) is a horizontal line crossing the y-axis at 2.
  • Example: \( y = -4 \) is a horizontal line crossing the y-axis at -4.

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Most popular questions from this chapter

Without graphing, do the following for each system of equations. (a) Describe each system. (b) State the number of solutions. (c) Is the system inconsistent, are the equations dependent, or neither? \(y+2 x=6\) \(x-3 y=-4\)

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A train travels \(150 \mathrm{~km}\) in the same time that a plane travels \(400 \mathrm{~km}\). If the rate of the plane is \(20 \mathrm{~km}\) per hr less than three times the rate of the train, find both rates.
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