Problem 9 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-12,\) find the slope and the \(y\) -intercept of the line with the given equation. $$y=10$$

Step-by-Step Solution

Answer

The slope of the line is 0 and the y-intercept is 10.

Step 1: Identify the form of the equation

The given equation is in the form \(y=k\), where \(k\) is a constant. This form represents a horizontal line.

Step 2: Find the slope of the line

Since the line is horizontal, the slope is 0. This is a property of horizontal lines.

Step 3: Find the y-intercept of the line

The y-intercept of a line given by the equation \(y=k\) is the value of \(k\). In this case, the y-intercept is 10.

Key Concepts

Horizontal Line Equation

When it comes to understanding lines on a graph, one of the simplest forms you'll encounter is the horizontal line equation. But what does this equation look like, and how does it behave on a graph? Imagine a line that runs straight across from left to right on your coordinate plane—this is a horizontal line.

Mathematically, a horizontal line has an equation that takes the form y = k, where k is a constant number. This number represents the precise vertical location of the line on the graph. No matter how far left or right you move along the line, the y-value stays consistent at k. This contrasts with vertical lines, which have a consistent x-value and are defined by an equation of the form x = k.

For example, the equation y = 10 represents a horizontal line that crosses the y-axis at 10. It is important to note that on such a line, all points have a y-value of 10, regardless of their x-value, indicating that the line is perfectly flat.

Slope of a Line

In mathematics, the 'slope' is a measure of the steepness or the inclination of a line. It is calculated as the ratio of the change in the y-value to the change in the x-value between two distinct points on a line, often represented as rise/run. If you were to walk along the line, the slope tells you how much you're going uphill or downhill.

To find the slope of a straight line between two points (x1, y1) and (x2, y2), you'd use the formula slope = (y2 - y1) / (x2 - x1). However, when dealing with a horizontal line like y = 10, there is no vertical change as you move along the line, because the y-value is constant; it always equals 10. As a result, the numerator of our slope formula, the rise, is zero, making the slope of a horizontal line 0. This is a distinctive feature that applies to all horizontal lines.

Y-Intercept

The y-intercept is where a line crosses the y-axis on a graph. To identify it, look for the point at which the x-value is zero. For a horizontal line like y = 10, the y-intercept is easily determined from the equation itself since the y-value doesn't change regardless of the x-value.

In fact, the constant k in the equation y = k represents the y-intercept directly. So for our example, the y-intercept is at the point (0, 10), meaning that the line crosses the y-axis 10 units above the origin. This concept is crucial for graphing linear equations and understanding the overall placement of a line in the coordinate system.