Problem 7 - Elementary Algebra

Question

For Problems 1-12, 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. $$ (3,-7), m=0 $$

Step-by-Step Solution

Answer

The standard form of the line equation is \(0x + 1y = -7\).

Step 1: Understanding the Information

We are given a point \((3, -7)\) and a slope \(m = 0\). With this information, we need to find the equation of a line in the form \(Ax + By = C\).

Step 2: Conceptualizing the Line Equation

Since the slope \(m\) is 0, the line is horizontal. The equation of a horizontal line is always \(y = c\), where \(c\) is the y-coordinate of any point on the line. Thus, the equation is \(y = -7\).

Step 3: Converting to Standard Form

To convert \(y = -7\) to the standard form \(Ax + By = C\):1. Recognize that \(x\) does not appear in the equation, so \(A=0\).2. The equation \(y = -7\) implies \(B = 1\) and \(C = -7\). Thus, we rewrite the equation as \(0 \, x + 1 \, y = -7\), simplifying to \(y = -7\).

Step 4: Verifying the Equation

Substitute the given point \((3, -7)\) into \(y = -7\). The point satisfies this equation because the y-coordinate is \(-7\). Hence, the equation is correct and expresses the line containing the point \((3, -7)\) with slope \(0\).

Key Concepts

Standard Form

In mathematics, the Standard Form of a linear equation is a way of expressing the equation to make it quite manageable and easy to work with. It is written in the format:
\[ Ax + By = C \]
This equation is particularly useful because both \(A\), \(B\), and \(C\) are integers, simplifying computational work and ensuring clarity. In this representation:

• \(A\) is the coefficient of \(x\).
• \(B\) is the coefficient of \(y\).
• \(C\) is the constant term on the other side of the equation.

For practical purposes, we often adjust equations into standard form to simplify integrating them with other equations.
For example, the equation of the exercise, originally in the format \(y = -7\), is transformed into \(0 \cdot x + 1 \cdot y = -7\), by recognizing that it does not involve \(x\), thus \(A = 0\) and \(B = 1\). This keeps the integrity of the line's characteristics intact while making it compatible with the standard form.

Slope of a Line

The slope of a line is a key attribute that provides understanding of its steepness or flatness. It is often represented as \(m\) in the equation of a line, indicating how fast the line rises or falls.
The slope is calculated by the ratio of **vertical change** to **horizontal change**:
\[ m = \frac{{\Delta y}}{{\Delta x}} \]

• A positive slope means the line rises from the left to the right.
• A negative slope indicates the line descends from the left to the right.
• A slope of zero implies a perfectly flat line, with no rise or fall.

In this exercise, the slope \(m = 0\), signifies that our line is horizontal. A slope of zero is unique because it assures that every point on the line has the same \(y\)-value, emphasizing the line's complete parallelism to the \(x\)-axis. Understanding slopes is essential for interpreting linear equations and their graphical representations.

Horizontal Line

A horizontal line is significant in geometry and algebra for its distinct properties.
It flows evenly across a plane, maintaining a constant \(y\)-coordinate no matter the \(x\)-value.
The equation of a horizontal line is expressed simply as \( y = c \), where \(c\) is the constant \(y\)-value showing its height on the plane.

• Every point on a horizontal line shares the same vertical position.
• It has a slope of zero, meaning there is no rise over run.
• These lines are always parallel to the \(x\)-axis.

In our exercise, we dealt with a horizontal line represented by \( y = -7 \). This means every point along this line has a \(y\)-coordinate of \(-7\). Whether you move left or right along the line, the height remains unchanged. Recognizing these traits helps when graphing and solving linear equations that represent horizontal lines.