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Chapter 2: Problem 16

Find the slope-intercept form of the line which passes through the given points. \(P(4,-8), Q(5,-8)\)

Short Answer

Expert verified
The slope-intercept form is \( y = -8 \).

Step by step solution

01

Identify the formula

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. We need to find both to express the equation in slope-intercept form.
02

Calculate the slope

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For points \( P(4,-8) \) and \( Q(5,-8) \), substitute the values in: \( m = \frac{-8 - (-8)}{5 - 4} = \frac{0}{1} = 0 \).
03

Determine the y-intercept

Since the slope \( m = 0 \), the line is horizontal and its y-intercept \( b \) is simply the y-value of the points, which is \( -8 \).
04

Write the equation in slope-intercept form

Putting together the slope \( m = 0 \) and the y-intercept \( b = -8 \), the equation is \( y = 0 \cdot x - 8 \). Simplifying this gives \( y = -8 \).

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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 represented by the letter \( m \), is a measure of how steep a line is. It tells us how many units the line moves up or down for each unit it moves horizontally. To calculate the slope between two points, \((x_1, y_1)\) and \((x_2, y_2)\), use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In simpler terms, the slope is the change in \( y \) divided by the change in \( x \). If you get a positive slope, the line is inclined upwards. A negative slope indicates a downward inclination. For a line where the slope is zero, the line is flat, meaning it is a horizontal line.
Y-Intercept
The y-intercept of a line is the point where the line crosses the y-axis. This value can be found using the slope-intercept form equation:
\( y = mx + b \)
Here, \(b\) is the y-intercept. It tells us where the line will cross the y-axis when the value of \(x\) is zero. Knowing the y-intercept helps us quickly understand one important characteristic of the line. In the case of a horizontal line, the y-intercept is always the same as the y-value of any point on the line, as the slope is zero.
Horizontal Line
A horizontal line is a straight line that runs from left to right. It does not rise or fall, which means its slope is zero. You can think of it as a perfectly flat line. For instance, if two points on the line have the same y-coordinate, like the points \((4, -8)\) and \((5, -8)\), the line is horizontal. These lines are represented by equations of the form:
\( y = b \)
The y-value remains constant, and the line extends infinitely in both horizontal directions.
Equation of a Line
The equation of a line in slope-intercept form is written as \( y = mx + b \). This form is extremely useful because it directly provides the slope \(m\) and the y-intercept \(b\). To write the equation of a line, determine its slope and y-intercept first. Then, plug these values into the formula.
  • The slope \(m\) indicates the ratio of rise over run between any two points on the line.
  • The intercept \(b\) shows where the line crosses the y-axis.

For the example of a horizontal line through the points \((4, -8)\) and \((5, -8)\), the equation simplifies to \( y = -8 \), since the slope \( m = 0 \) and the intercept \( b \) is -8. This simplicity is characteristic of horizontal lines.

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