Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). With \(y\)-intercept 5 and slope 0

Step-by-Step Solution

Answer

The equation is \( y - 5 = 0 \).

Step 1: Identify Line Characteristics

The problem states that the line has a slope of 0 and a y-intercept of 5. A slope of 0 means the line is horizontal.

Step 2: Formulate the Equation

For a horizontal line with a y-intercept, the equation is simply the constant value of the y-intercept. Therefore, the equation is given by \( y = 5 \).

Step 3: Convert to the Standard Form

The standard form of a line is \( Ax + By + C = 0 \). For the equation \( y = 5 \), we have \( 0x + 1y - 5 = 0 \). This simplifies to \( y - 5 = 0 \).

Key Concepts

Slope

The concept of slope is central in understanding linear equations. Think of the slope as a measure of the steepness or incline of a line. It tells us how much the line rises or falls as we move from one point to another.
When dealing with a slope of 0, as in this problem, the line is horizontal. This means that no matter how far you go along the x-axis, the y-value remains constant. In mathematical terms, a horizontal line has a slope of 0 because there is no change in the y-values as the x-values change.

The slope is calculated as the change in the vertical direction (rise) over the change in the horizontal direction (run). For a line with slope 0:

• There is no rise, the vertical change is 0.
• This means the slope formula, given by \( m = \frac{\Delta y}{\Delta x} \), simplifies to \( m = 0 \).

This concept helps us understand the nature of lines and how we can predict their behavior graphically.

Y-intercept

The y-intercept is where the line crosses the y-axis in a coordinate plane.
In a linear equation, the y-intercept is represented by the constant term. It provides a starting point for the line. In the equation form \( y = mx + b \), the y-intercept is \( b \).

In the given problem, the y-intercept is 5. This means that the line crosses the y-axis at 5. For horizontal lines, the y-intercept is the only value of \( y \) throughout the line, which simplifies the representation to just \( y = b \). In this case,
the equation becomes \( y = 5 \).
By understanding the y-intercept, it becomes easier to graph the equation and see that the line remains constant at this value.

Standard Form of a Line

The standard form of a linear equation is another way to express the equation of a line. It is written as \( Ax + By + C = 0 \). Here, \( A \), \( B \), and \( C \) are constants, and \( x \) and \( y \) are variables representing any point on the line.

To convert the equation \( y = 5 \) into standard form:

• Recognize that there is no \( x \) term, implying \( A = 0 \).
• The \( y \) term has a coefficient of 1 (\( B = 1 \)).
• The constant term is -5 (\( C = -5 \)), leading to the equation \( 0x + 1y - 5 = 0 \).

This provides structure and is widely used for linear programming and systems of equations.
It helps in easily identifying parallel and perpendicular lines by examining the coefficients \( A \) and \( B \). Understanding the standard form is crucial for deeper algebraic analysis and solving more complex equation systems.