Problem 9 - Introductory Algebra for College Students

Question

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 given by the equation \(y = 10\) is 0 and the y-intercept is 10.

Step 1: Identify the Slope

Since there is no 'x' term in the equation, this indicates that the slope of the given line is 0. In the slope-intercept form of a line, the coefficient of the 'x' term denotes the slope. The absence of an 'x' term represents a slope of zero which is characteristic of horizontal lines.

Step 2: Identify the y-Intercept

The y-intercept \(b\) is the coefficient of the 'y' term in the given equation. In the example, the y-intercept is 10.

Key Concepts

Understanding the Slope

The concept of slope is crucial in understanding the direction and steepness of a line in a graph. Slope is expressed in the slope-intercept form of a linear equation as the coefficient of the 'x' term.For example, in an equation like \( y = mx + b \), \( m \) is the slope. It tells us how much the 'y' value changes for a unit change in 'x'. A higher slope value will indicate steepness, and a negative slope will indicate that the line goes down as you move to the right on a graph.However, in the case of a horizontal line like \( y = 10 \), the slope is 0. This means there is no rise or fall; the line is flat and parallel to the x-axis. Any changes in 'x' do not impact the 'y' value.

Exploring the Y-Intercept

The y-intercept of a line is where the line crosses the y-axis. It is represented by \( b \) in the slope-intercept equation \( y = mx + b \). The y-intercept is a specific point that can help in plotting the graph of a line, showing where the line sits on the y-axis when \( x = 0 \). For a line with the equation \( y = 10 \), the y-intercept is 10. This means the line crosses the y-axis at the point (0,10). It provides insight into the vertical position of the entire line.

Horizontal Lines Explained

Horizontal lines are unique in that they are completely flat and stretch left to right. They are parallel to the x-axis and are easily identifiable through their equation.When you see an equation like \( y = 10 \), it describes a horizontal line. In these equations, the absence of an 'x' term signals that the 'y' value remains constant. Such a line will always have a slope of 0, indicating no vertical change as 'x' increases or decreases.Some important points regarding horizontal lines include:

• The y-intercept is the constant value of 'y', which all points on the line share.
• They do not cross the x-axis unless \( y = 0 \).
• They're often used in real-life applications where a quantity remains unchanged over time.

Horizontal lines simplify understanding, as they represent constancy and lack of incline.