Problem 30 - Algebra A Combined Function

Question

Write an equation of each line. Write the equation in the form \(x=a, y=b\), or \(y=m x+b\). See Examples 5 and \(6 .\) Through (-2,6)\(;\) perpendicular to \(y=7\)

Step-by-Step Solution

Answer

The equation of the line is \(x = -2\).

Step 1: Understand Line Characteristics

The given line is represented by the equation \(y = 7\), which is a horizontal line. All horizontal lines have a slope of 0.

Step 2: Identify the Slope of Perpendicular Line

Lines that are perpendicular to horizontal lines are vertical lines. Vertical lines have an undefined slope and are written in the form \(x = a\), where \(a\) is the x-coordinate of any point on the line.

Step 3: Formulate the Equation of the Perpendicular Line

Since the line must pass through the point (-2, 6) and be perpendicular to \(y = 7\), it must be vertical and take the form \(x = -2\).

Step 4: Verify the Line Equation

Check that the line \(x = -2\) passes through the point (-2, 6). Since the x-coordinate of the line and the point are the same, the line indeed passes through (-2, 6).

Key Concepts

Perpendicular Lines

Perpendicular lines are quite special in geometry and algebra. They intersect at a right angle, which is 90 degrees. This relationship between lines involves their slopes. If one line is horizontal, the perpendicular line will be vertical, and vice versa.

• Horizontal lines have a slope of 0.
• Vertical lines have an undefined slope.

Understanding this concept is crucial when working with linear equations. If you know the slope of one line and need to find a perpendicular line, you must consider this complementary relationship.

So, when asked to find a line that is perpendicular to another, always think about their intersecting nature and their slopes.

Vertical Lines

Vertical lines run straight up and down. They have a distinct characteristic in that their slope is undefined. This happens because slope is calculated with the formula \( m = \frac{\Delta y}{\Delta x} \), and for vertical lines, \( \Delta x = 0 \). Division by zero is undefined, which gives these lines their unique property.

Vertical lines are expressed in the form \( x = a \), where \( a \) is the x-coordinate through which the line passes. Since it never moves horizontally, its x-value remains constant no matter what point on the line you examine.

For example, a vertical line passing through the point \((-2, 6)\) will simply be \( x = -2 \). It doesn’t matter what the y-coordinate is. So anytime you're dealing with a perpendicular line to a horizontal one, it's this concept of vertical lines that comes into play.

Slope

The slope of a line is a critical concept in geometry. It's a measure of how steep the line is. Calculated by the formula \( m = \frac{\Delta y}{\Delta x} \), the slope tells us the change in \( y \) (vertical) against the change in \( x \) (horizontal).

• Positive slopes rise as you move from left to right.
• Negative slopes fall as you move from left to right.
• A slope of 0 indicates a horizontal line.

Slope is fundamental in understanding the orientation of lines. It's especially important when identifying perpendicular lines, as the product of the slopes of two perpendicular lines is -1. However, in some cases, such as when dealing with horizontal and vertical lines, one slope will be zero, and the other will be undefined, reinforcing their perpendicular nature without calculation.