Problem 10 - 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. $$y=7$$

Step-by-Step Solution

Answer

The slope of the line is 0 and the y-intercept is 7.

Step 1: Identify the slope

In the given equation \(y = 7\), there is no \(x\) term. This indicates that the slope \(m\) of the line is 0.

Step 2: Identify the y-intercept

In the given equation \(y = 7\), the term 7 is the y-coordinate where the line intersects the y-axis, hence the y-intercept \(b\) of the line is 7.

Key Concepts

Slope

The slope of a line is a key concept in algebra. It tells us how steep a line is. Think of slope as the tilt or incline of a line on a graph. Mathematically, slope is represented by the letter \( m \) and is defined as the ratio of the "rise" (vertical change) over the "run" (horizontal change) between two points on a line. This can be expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• "Rise" refers to the change in the y-values, or the vertical distance.
• "Run" refers to the change in the x-values, or the horizontal distance.

In the equation \( y = 7 \), we observe that there is no \( x \) term present. This means the line has a slope of 0. A slope of 0 means the line is completely flat, indicating no vertical change as \( x \) moves left or right.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is a fundamental aspect of the equation of a line in the form \( y = mx + b \). Here, \( b \) is the y-intercept. It tells us the exact point at which the line enters or touches the y-axis.

• The y-intercept is always of the form \( (0, b) \), indicating that the x-value at the point of intersection on the y-axis is zero.
• For any line, knowing the y-intercept helps in sketching the line on a coordinate plane.

In the given equation \( y = 7 \), the line crosses the y-axis at the point \( (0, 7) \). Therefore, the y-intercept is 7. This tells us the value of \( y \) when \( x \) is 0.

Horizontal Line

A horizontal line is one that remains at a constant y-value across all x-values. It's like a path drawn straight across the graph without any incline or decline. In the context of the equation \( y = 7 \), this line is horizontal, suggesting that no matter what value \( x \) takes, \( y \) will always be 7.

• Horizontal lines can be identified by equations where the y-value is constant and there is no x-term, such as \( y = c \), where \( c \) is a constant.
• The slope of a horizontal line is always 0, reflecting its lack of vertical movement.

These lines are parallel to the x-axis and represent scenarios where there is no change in y, regardless of x. For instance, the line \( y = 7 \) appears as a flat, horizontal line at y = 7, perfectly parallel to the x-axis.