Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine the slope and y-intercept (if possible) of the linear equation. Then describe its graph. $$y=12$$

Step-by-Step Solution

Answer

The slope of the given equation is 0 and the y-intercept is 12. The graph of the equation is a horizontal line parallel to x-axis that intersects the y-axis at the point (0,12).

Step 1: Determine the Slope

For the equation of the form \(y=b\), where \(b\) is a constant, the slope is always 0. So, the slope of the given equation \(y=12\) is 0.

Step 2: Determine the y-intercept

For the equation of the form \(y=b\), where \(b\) is a constant, the y-intercept is the value of \(b\). So, the y-intercept of the equation \(y=12\) is 12.

Step 3: Describe the Graph

The graph of the equation \(y=12\) is a horizontal line parallel to x-axis that intersects the y-axis at the point (0,12). Since the slope is 0, there's no inclination and the line remains flat, parallel to the x-axis.

Key Concepts

Understanding Slope

In the world of linear equations, the slope is a key concept that describes the direction and steepness of a line. Simply put, the slope tells us how much the line rises or falls as it moves horizontally across the graph. For any equation in the form \(y = b\), such as our equation \(y=12\), the slope is 0. This is because as you move along the line, the y-value does not change—it remains constant at 12, no matter the x-value. A slope of 0 indicates that the line is completely horizontal. This means it does not ascend or descend at all, staying perfectly flat across the graph.

Identifying the Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In linear equations, it's expressed as \(b\) in the form \(y = mx + b\). But when it comes to equations like \(y = 12\), where there is no \(m\) or \(x\), the concept simplifies. The given constant, 12, directly tells us the y-intercept. Hence, for \(y = 12\), the line intercepts the y-axis at 12. This means that when \(x\) is 0, \(y\) is 12, giving us the exact point of intersection at (0, 12).

What is a Horizontal Line?

A horizontal line is characterized by having a slope of 0. This type of line runs parallel to the x-axis.
Horizontal lines like \(y = 12\) have unique features:

• They do not rise or fall, ensuring that the y-value remains consistent across all x-values.
• The equation is independent of x, meaning no matter what value x takes, y will always be 12.
• Such lines are always flat, with no tilt towards the x-axis.

In real-world scenarios, horizontal lines can represent situations where a specific condition remains constant, such as a flat rate or a fixed salary.

Describing the Graph

Graphically interpreting the equation \(y = 12\) provides a clear visual representation that is easy to understand. Upon plotting the equation, the line appears horizontally and traverses from left to right, never changing its y-value of 12. Here's how we can describe this graph:

• The line runs parallel to the x-axis, illustrating it is horizontal.
• It intersects the y-axis precisely at the point (0, 12), corresponding to the y-intercept.
• Due to the zero slope, the line lacks any inclination or decline.

This comprehensive description makes it evident that the equation results in a graph where, no matter the x-value, the y-value steadfastly remains 12, embodying a perfect horizontal line that maintains its steadiness across the graph.