Problem 52 - Elementary Algebra

Question

For Problems \(45-60\), write the equation of the line that satisfies the given conditions. Express final equations in standard form. Contains the point \((-4,7)\) and is perpendicular to the \(x\) axis

Step-by-Step Solution

Answer

The line is given by the equation: \(x = -4\).

Step 1: Understanding Perpendicularity to the x-axis

Lines that are perpendicular to the x-axis are vertical lines. Vertical lines have an undefined slope because they go straight up and down.

Step 2: Identifying the Equation of a Vertical Line

A vertical line has the equation of the form \(x = a\), where \(a\) is the constant x-value for every point on the line.

Step 3: Substitute the Given Point

Since the line must pass through the point \((-4, 7)\), substitute \(-4\) for \(a\) in the vertical line equation. The equation becomes \(x = -4\).

Step 4: Expressing in Standard Form

The standard form for a line is \(Ax + By = C\). For a vertical line, \(B = 0\). Rewrite the equation \(x = -4\) in standard form: \(1x + 0y = -4\), or simply \(x = -4\).

Key Concepts

Perpendicular Lines

When two lines are perpendicular, they intersect at a right angle (90 degrees). You can often find these lines in various geometric shapes and their understanding is crucial in many mathematical problems. In coordinate geometry, if a line is perpendicular to the x-axis, it is a vertical line. Knowing how to work with perpendicular lines includes:

• Understanding that perpendicular slopes multiply to -1, but this doesn't apply when one of the lines is vertical or horizontal.
• Realizing that the concept "perpendicular to the x-axis" translates to a vertical line.
• Using this knowledge to identify perpendicular relationships, specifically vertical lines that run parallel to the y-axis.

Perpendicular lines are commonly used in construction, graphing, and designing when we need precise angles.

Vertical Lines

A vertical line is a straight line that runs up and down the graph. These lines are unique because they are described by equations of the form, \( x = a \). Here, "a" represents the x-coordinate through which the line passes. In our exercise, the line passes through the x-coordinate of -4, thus the equation is \( x = -4 \). Key features of vertical lines include:

• Having an undefined slope, which makes them perpendicular to any horizontal line like the x-axis, which has a zero slope.
• Being parallel to the y-axis.
• Not having a y-intercept because they never cross the y-axis.

Vertical lines are important for dividing the graph into sections and helping in visualizing data aligned along the y-axis.

Standard Form

The standard form of the equation of a line is a way of storing an equation that makes it easy to understand and work with. It is expressed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers. In the context of our problem, the line \( x = -4 \) is already almost in standard form.To convert a vertical line equation into standard form, note:

• The coefficient \( A \) corresponds to the x-term, and \( B \) corresponds to the y-term.
• For vertical lines, \( B \) is always 0, because the line is parallel to the y-axis and does not have a y-component.
• Thus, the equation \( x = -4 \) can be converted to \( 1x + 0y = -4 \).

The standard form is frequently used because it allows for the easy comparison of different lines and can be very useful in solving system of equations problems.