Problem 27 - Precalculus

Question

Find an equation for the line that is described, and sketch the graph. Write the answer in the form \(A x+B y+C=0\). Passes through (-3,4) and is parallel to the \(x\) -axis.

Step-by-Step Solution

Answer

The equation is \( 0x + 1y - 4 = 0 \). It's a horizontal line at \( y = 4 \).

Step 1: Understand Characteristics of the Line

The problem states that the line is parallel to the x-axis, meaning the slope of the line is 0. A line parallel to the x-axis has the form \( y = c \), where \( c \) is a constant value.

Step 2: Determine the Equation of the Line

Since the line is parallel to the x-axis and passes through the point (-3,4), we can use the point to find \( c \). Substitute \( y = 4 \) because when a line is horizontal (parallel to the x-axis), the y-value is constant. Thus, the equation of the line is \( y = 4 \).

Step 3: Convert the Equation to the Standard Form

To express the equation \( y = 4 \) in the form \( Ax + By + C = 0 \), we rewrite it as \( 0x + 1y - 4 = 0 \), which simplifies to \( y - 4 = 0 \). Hence, the equation in the standard form is \( 0x + 1y - 4 = 0 \).

Key Concepts

Parallel to the x-axis

Lines that run parallel to the x-axis have a unique characteristic—they never intersect with the x-axis. These lines are perfectly horizontal. A horizontal line corresponds to a situation where there is no change in the y-coordinate as you move along the x-axis. This means that the line has a constant y-value, regardless of the x-value.
For a line parallel to the x-axis, its equation can be written simply as:

• \( y = c \), where \( c \) is a constant

The constant \( c \) represents the y-coordinate of every point on the line. For instance, if a line is parallel to the x-axis and passes through the point (-3, 4), then \( c = 4 \). Therefore, the equation of the line is \( y = 4 \). This line will remain at this y-level across all x-values, illustrating its horizontal nature.

Standard form equation

When dealing with lines in geometry, expressing an equation in the standard form is a common practice. The standard form of a line's equation is represented as:

• \( Ax + By + C = 0 \)

Here, \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) should not both be zero. The goal of converting an equation into this form is to provide a consistent and recognizable structure that can ease the analysis and graphing of the line.
In the case of a line that is parallel to the x-axis, the equation generally starts as \( y = c \). To convert this into standard form:

• Recognize that \( y = c \) implies \( 0x + 1y - c = 0 \).

Using the example from the exercise, \( y = 4 \) becomes \( 0x + 1y - 4 = 0 \), or simply \( y - 4 = 0 \). This aligns the equation into the standard form, making it easy to identify and work with.

Slope of a line

The slope of a line is a measure of its steepness and is usually represented by the letter \( m \). It tells us how much the y-coordinate of a point on the line changes as the x-coordinate changes.

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope means a vertical line.

For lines parallel to the x-axis, like the one in our exercise, the slope is 0. This is because there is no vertical change as you move along the line—it's completely flat. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

However, for a line like \( y = 4 \), since both \( y_1 \) and \( y_2 \) would equal 4, the equation becomes \( m = \frac{4 - 4}{x_2 - x_1} = 0 \). Thus, confirming the line is horizontal and parallel to the x-axis.