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

Short answer

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

Step-by-step solution

Identifying Slope and Y-Intercept

For this problem, we are given that the line has a slope of 0 and a y-intercept of 5. A line with a slope of 0 is horizontal.

Setting Up Slope-Intercept Form

The general form of a line in slope-intercept form is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 0 \) and \( b = 5 \), so we have \( y = 0 \cdot x + 5 \).

Simplifying the Equation

Simplify \( y = 0 \cdot x + 5 \) to \( y = 5 \). This is the equation of the line in slope-intercept form.

Converting to Standard Form

To write \( y = 5 \) in standard form \( Ax + By + C = 0 \), notice that there is no \( x \) term, so we can write it as \( 0 \cdot x + 1 \cdot y - 5 = 0 \), which simplifies to \( y - 5 = 0 \).

Key concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way of expressing the equation of a line. It's given by the formula \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis.
For instance, if you know the slope of the line and the y-intercept, you can easily substitute these values into the formula to get the equation of the line. For a horizontal line with a slope of 0, the equation simplifies significantly because 0 times any number is 0. So, if we have a y-intercept at 5, the slope-intercept form becomes simply \( y = 5 \).
Using the slope-intercept form makes it easy to graph the line or predict its behavior. Set any value of \( x \), and you'll find a consistent \( y \), making plotting a breeze.
**Key Benefits:**

• Straightforward to understand and use.
• Quickly shows the y-intercept and slope.
• Makes graphing simple and intuitive.

Standard Form

The standard form of a line's equation is another way to express it, typically written as \( Ax + By + C = 0 \). Here, \( A \), \( B \), and \( C \) are integers, and \( A \) should usually be positive.
This form is particularly useful in specific mathematical contexts, such as analyzing and solving systems of equations. In the given exercise, the horizontal line \( y = 5 \) can be rewritten in standard form. Since it does not have an \( x \) term, the equation can be formatted as \( 0 \, x + 1 \, y - 5 = 0 \), which simplifies to \( y - 5 = 0 \).
While it might look more complicated initially, converting equations to standard form can allow for easier manipulation in some calculations. For example, it can help when computing intersections with other lines or applying linear algebra.
**Considerations When Using Standard Form:**

• Always aim for integer values in the equation.
• Useful for solving simultaneous equations.
• Can be less intuitive for direct graphing purposes.

Slope

The slope of a line essentially measures how steep the line is. It's often denoted by \( m \) in the slope-intercept form, \( y = mx + b \).
The slope depicts the ratio of the rise (change in \( y \)) to the run (change in \( x \)). In essence, it tells us how much \( y \) changes for a unit change in \( x \). A positive slope indicates an upward trend, a negative slope a downward trend, while a slope of 0 signifies a horizontal line.
In our exercise, a slope of 0 implies there's no vertical change as \( x \) varies, keeping the line flat. For example, \( y = 5 \) remains constant and doesn't angle up or down.
**Understanding Slope Helps You To:**

• Determine the line's direction.
• Anticipate the line's behavior over changes in \( x \).
• Easily identify horizontal or vertical lines (0 or undefined slope).

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's found by setting \( x = 0 \) in the line's equation, making it straightforward to determine from the equation in slope-intercept form \( y = mx + b \). Here, \( b \) directly represents the y-intercept.
For the exercise at hand, the y-intercept is 5. That tells us that when \( x = 0 \), \( y = 5 \), so the line crosses the y-axis at the coordinate (0, 5).
The y-intercept provides a crucial point for graphing a line. It's often one of the first points plotted when drawing lines manually or via software tools.
**Significance of the Y-Intercept:**

• Immediately shows the starting point for the line on the y-axis.
• Integral in constructing the line's graph manually.
• Offers easy insight into the line's position related to the y-axis.