Problem 50 - Elementary Algebra

Question

Write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((-3,-7)\) and is parallel to the \(x\) axis

Step-by-Step Solution

Answer

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

Step 1: Understand the Properties of Lines Parallel to the x-axis

Lines that are parallel to the x-axis have a slope of zero. This means that the y-value does not change, regardless of the x-value.

Step 2: Identify the Y-Value for the Line

Given the point (-3, -7) that lies on the line, the y-value of every point on this line must be -7 because the line is parallel to the x-axis.

Step 3: Write the Equation in Point-Slope Form

For lines parallel to the x-axis, the equation that represents all such lines is \(y = c\), where \(c\) is a constant. Since the line contains the point (-3, -7), the equation becomes \(y = -7\).

Step 4: Express the Equation in Standard Form

The standard form of a linear equation is \(Ax + By = C\). For the equation \(y = -7\), we can rewrite it as \(0x + y = -7\). This is in the form of \(Ax + By = C\), where \(A = 0\), \(B = 1\), and \(C = -7\).

Key Concepts

Parallel Lines

Parallel lines in geometry are fascinating because they never intersect. This characteristic defines them as they share the same direction or slope. Hence, they always maintain a constant distance from one another.

• When analyzing parallel lines in the context of linear equations, especially in a coordinate plane, their slopes play a critical role.
• The slope, often denoted as "m" in equations, must be the same for lines to be parallel.

For example, any line parallel to the x-axis will have a slope of zero. This is because the line stays perfectly horizontal, and the y-values do not alter regardless of changes in the x-values. This means the rate of change, which is the slope, remains zero, illustrating a horizontal parallel line.
Understanding these concepts deeply can elevate your geometry skills, as identifying parallel lines is a common requirement in many math problems and real-life applications.

Standard Form

The standard form of a linear equation is a neat way to represent lines and is known for its clear structure. In this form, an equation is organized as:

• \[Ax + By = C\]

Here, \(A\), \(B\), and \(C\) are integers, where \(A\) and \(B\) are not both zero. Some guidelines to remember:

• \(A\) should ideally be a positive integer.
• The equation often gets adjusted to avoid fractions and decimals, enhancing clarity.

Expressing a line in standard form is beneficial as it can quickly reveal the line's vertical inclination or horizontal nature. For instance, when a line is parallel to the x-axis, this becomes evident since \(A = 0\), translating to an emphasis on the y-values alone (e.g., \(0x + y = C\)).
Grasping the conversion to standard form can aid in solving and visualizing linear equations more effectively.

Slope of a Line

The slope of a line is a powerful concept in coordinate geometry that provides insight into the direction and steepness of a line. The slope represents the rate at which the y-coordinate changes with respect to the change in the x-coordinate. Often denoted as \(m\), it is calculated using the formula:

• \[m = \frac{y_2 - y_1}{x_2 - x_1}\],

where \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on a line.

• A positive slope indicates the line ascends (goes up left to right).
• A negative slope means the line descends (goes down left to right).
• A slope of zero is characteristic of horizontal lines (parallel to the x-axis), signaling no change in the y-values.
• An undefined slope is typical of vertical lines, indicating no change in x-values.

In our example, the line described is horizontal, with a slope of zero, illustrating zero change in the vertical direction. Understanding slope helps in predicting the behavior of linear equations and plays a pivotal role in graphing.