Problem 44 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. With slope \(0,\) through (6.7,12.1)

Step-by-Step Solution

Answer

The equation of the line is \(y = 12.1\).

Step 1: Understand the Problem

The problem asks us to find the equation of a line that has a slope of zero and passes through the point (6.7, 12.1). We need to put the equation in slope-intercept form, which is usually expressed as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Recognize the Slope of the Line

A slope of 0 indicates that the line is horizontal. This means that the y-value remains constant across all x-values along the line.

Step 3: Write the Equation using the Slope

Since the line is horizontal with a slope of 0, the equation of the line will be of the form \(y = b\). We need to find the value of \(b\).

Step 4: Determine the y-intercept

Given that the line passes through the point (6.7, 12.1), this means that the value of \(b\), the y-intercept, is 12.1. Hence, the equation of the line is \(y = 12.1\).

Key Concepts

Equation of a Line

When we talk about the equation of a line, we are usually referring to its representation in a particular mathematical form. A common and useful format is the slope-intercept form, expressed as \( y = mx + b \). In this form, \( m \) represents the slope of the line, while \( b \) signifies the y-intercept.

• The slope \( m \) tells us how steep the line is. It indicates how much the y-coordinate of a point on the line increases (or decreases) as the x-coordinate increases by one unit.
• The y-intercept \( b \) is the point where the line crosses the y-axis. It tells us the value of \( y \) when \( x \) is zero.

To write the equation of any line, you need to determine these two variables. Once you have \( m \) and \( b \), you can substitute these values into the formula to find the line's equation. In the case where the slope \( m \) is zero, the equation simplifies significantly, highlighting another interesting concept: horizontal lines.

Horizontal Line

A horizontal line is a special kind of line that runs parallel to the x-axis. One of its main characteristics is that it has a slope of zero. This means that no matter how much you move along the x-axis, the y-value remains unchanged.

• For a horizontal line, the equation takes a simple form: \( y = b \).
• The value of \( b \) is the constant y-coordinate of all points on the line.

For example, consider a line that passes through the point \( (6.7, 12.1) \) and is horizontal. This means every point on this line has the y-coordinate 12.1. Regardless of what the x-value is, \( y = 12.1 \) throughout. This constancy is what gives horizontal lines their straightforward and simple equation. It's essential to remember that these lines maintain a fixed y-value and do not rise or fall.

Y-Intercept

The y-intercept is a fundamental concept in understanding graphs of equations and lines. It is the point where a line crosses the y-axis, and thus, it's the value of \( y \) when \( x = 0 \).

• In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
• It's essential for forming the equation of a line, as it helps pinpoint the line's starting or reference point on the graph.

In horizontal lines, like in our example of \( y = 12.1 \), the y-intercept is also the constant value of \( y \) for all points. Because horizontal lines run parallel to the x-axis and do not intersect it, they essentially have a y-intercept for every value of \( x \). Understanding the y-intercept helps in sketching the line accurately and predicting its behavior on a graph.