Problem 33 - College Algebra

Question

For the following exercises, find the equation of the line using the given information. The slope equals zero and it passes through the point \((1,-4) .\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -4 \).

Step 1: Understand the Properties

We are given that the slope of the line is zero. A line with a slope of zero is a horizontal line. This means that the line does not rise or fall as it moves along the x-axis.

Step 2: Identify the Equation of a Horizontal Line

Since the slope is zero, the equation of a horizontal line is given by the form \( y = b \), where \( b \) is the constant y-value for any point on the line.

Step 3: Use the Given Point to Determine \( b \)

We know that the line passes through the point \((1, -4)\). Since it is a horizontal line, the y-coordinate of this point is \( -4 \). Therefore, \( b = -4 \).

Step 4: Write the Equation of the Line

Substitute \( b = -4 \) into the horizontal line equation. So, the equation of the line is \( y = -4 \).

Key Concepts

Understanding Slope

Slope is a fundamental concept when dealing with lines in algebra. It's a measure of how steep a line is. Think of walking up a hill: the steeper the hill, the greater the slope.

The slope is calculated by the change in the y-values divided by the change in the x-values within a given interval. We often use the formula \( m = \frac{\Delta y}{\Delta x} \), where \( m \) is the slope. A positive slope means the line goes upwards as it moves from left to right. A negative slope suggests it goes downwards.

However, there are certain special cases, such as when the slope is zero. A zero slope means the line is perfectly flat and runs parallel to the x-axis. This characteristic is vital in recognizing horizontal lines.

Characteristics of a Horizontal Line

A horizontal line is one that stretches left to right without any upward or downward tilt. These lines are unique because their slope is always zero.

Since a horizontal line never climbs or descends, it always maintains the same y-coordinate value for any given x-coordinate. For example, if you have a horizontal line that passes through the point \((1, -4)\), every point along that line will share the y-coordinate of -4 while x can take any value.

This specific trait is crucial in helping us easily identify horizontal lines in equations and graphs.

The Role of Y-Coordinate

When discussing lines, especially horizontal ones, the y-coordinate becomes a primary focus. The y-coordinate tells us exactly where the line crosses the y-axis.

For horizontal lines, the y-coordinate is always the same, regardless of the x-coordinate. If a horizontal line passes through a point with a y-coordinate of -4, it means every point on the line has this value, making it an essential component to define the equation of the line.

Knowing the y-coordinate is not just about pinpointing a spot; it helps us write the line's equation and fully grasp what the line looks like on a graph.

Determining the Equation of a Line

The equation of a line provides a clear mathematical description of every point along the line. For a horizontal line, this equation takes a simple form: \( y = b \), where \( b \) is the constant y-coordinate.

In our scenario, the line passes through the point \((1, -4)\). This tells us directly that \( b = -4 \), so the equation of the line is \( y = -4 \).

This equation means no matter which x-value you choose, the y-coordinate is always -4, reinforcing the idea of a horizontal, flat line. Understanding how to derive this equation from a set of characteristics is key in mastering linear equations.