Problem 7 - Elementary Algebra

Question

Find the equation of the line that contains the given point and has the given slope. Express equations in the form \(A x+B y=C\), where \(A, B\), and \(C\) are integers. (Objective 1a) $$(3,-7), m=0$$

Step-by-Step Solution

Answer

The equation is \(0x + 1y = -7\).

Step 1: Identify Given Information

The problem provides a point \((3, -7)\) and a slope \(m = 0\). This means we need to find the equation of a horizontal line that passes through the given point.

Step 2: Understand Horizontal Line Equation

For a horizontal line, all points on the line have the same y-coordinate because the slope is zero. Therefore, the equation of a horizontal line is of the form \(y = c\), where \(c\) is the y-coordinate of the given point.

Step 3: Write the Equation in Slope-Intercept Form

Using the point \((3, -7)\), the equation of the horizontal line is \(y = -7\).

Step 4: Convert to Standard Form

The standard form for a line's equation is \(Ax + By = C\). Here, the equation \(y = -7\) can be rewritten as \(0x + 1y = -7\), where \(A = 0\), \(B = 1\), and \(C = -7\).

Key Concepts

Slope-Intercept Form

When tackling linear equations, the slope-intercept form is a popular choice because of its straightforward nature. It's represented as \( y = mx + b \), where \( m \) stands for the slope and \( b \) for the y-intercept—the point where the line crosses the y-axis. This form is especially handy when you know these two pieces of information.

In the given exercise, the slope \( m \) is 0, indicating a flat, horizontal line. The slope value tells us how steep the line is; in this case, no steepness means the line won't rise or fall as you move along the x-axis. The y-intercept, \( -7 \), tells us exactly where the line will meet the y-axis. Together, these factors compose the equation \( y = -7 \) in slope-intercept form, providing a clear picture of the line's behavior.

Some advantages of using this form include:

• Quickly identifying the slope and y-intercept from the equation.
• Ease of graphing the line on a coordinate grid.

Standard Form

Another way to express linear equations is through the standard form, noted as \( Ax + By = C \), where \( A, B, \) and \( C \) are integers, and importantly, \( A \) should not be negative. This form is common in algebra as it clearly shows the relationship between x and y, allowing for simplification and easy comparison between multiple lines.

In our exercise, we transition from the slope-intercept form \( y = -7 \) to the standard form \( 0x + 1y = -7 \). Here, \( A = 0 \) and \( B = 1 \), which highlights that there is no x-component contributing to the equation. This is typical for horizontal lines.

Some benefits of the standard form include:

• The ability to easily perform algebraic operations like addition or subtraction with multiple equations.
• Clarity in highlighting the linear relationship between x and y, especially when dealing with contexts involving integers.

Horizontal Lines

Understanding the characteristics of horizontal lines in linear equations is crucial since they present a unique case. A horizontal line has a slope of zero, meaning it does not rise or fall—it remains perfectly level as you move left or right. Because of this, the equation of a horizontal line is simple and often takes the form \( y = c \), where \( c \) is constant.

In the exercise, the line passes through the point \((3, -7)\), so all points on this line share the same y-coordinate: -7. The x-value can be any number, but the y-value remains fixed at -7, giving us the simple equation \( y = -7 \).

Horizontal lines are easy to spot on a graph because they run parallel to the x-axis and are defined solely by their y-coordinate.