Problem 32 - Precalculus

Question

Are the lines \(y=x+1\) and \(y=1-x\) parallel, perpendicular, or neither?

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Identifying Slope

The first step is to identify the slope of each line. For the line equation in the form of \(y = mx + c\), \(m\) represents the slope. For the line \(y = x + 1\), the slope \(m_1\) is 1, as the coefficient of \(x\) is 1. For the line \(y = 1 - x\), rewriting it as \(y = -x + 1\), the slope \(m_2\) is -1, since the coefficient of \(x\) is -1.

Step 2: Checking Parallelism

Lines are parallel if they have the same slope. Here, the slopes are \(m_1 = 1\) and \(m_2 = -1\). Since these slopes are not equal, the lines are not parallel.

Step 3: Checking Perpendicularity

Lines are perpendicular if the product of their slopes is -1. Calculate the product of slopes: \(m_1 \times m_2 = 1 \times (-1) = -1\). Since the product is -1, the lines are perpendicular.

Key Concepts

Parallel Lines

Parallel lines are fascinating because they always maintain the same distance from each other and never intersect. They can run side by side indefinitely without meeting. This property is possible primarily due to their identical slopes.To determine whether two lines are parallel, check their slopes:

• If the slopes are equal, the lines are parallel.
• If the slopes are different, the lines are not parallel.

For instance, consider two line equations like the ones in the original exercise. If both line equations are in the slope-intercept form (\( y = mx + c \)), and if the slopes \( m \) are the same, then these two lines will indeed be parallel. Remember, having the same \( y \text{-intercept} \) is not necessary for lines to be parallel; only the slopes need to match!

Perpendicular Lines

Perpendicular lines are quite special due to their mutual angles. At their intersection, they form a "T" shape, which splits into four right angles. What truly defines them mathematically is the relationship between their slopes.To verify if lines are perpendicular:

• The product of their slopes should be \(-1\).
• If the product is \(-1\), the two lines are exactly perpendicular.

Let's dive into the lines from the exercise: one with slope \( m_1 = 1 \), and another with slope \( m_2 = -1 \). When multiplied, the slopes give \( 1 \times (-1) = -1 \). This confirms that these lines perfectly intersect at right angles, making them perpendicular. Never forget this neat trick for quickly identifying perpendicular lines: flip the reciprocal and change the sign. Hence, a slope of \( 2 \) would need \(-\frac{1}{2}\) to be its perpendicular pair!

Line Equations

Understanding line equations is crucial for mastering coordinate geometry. The slope-intercept form, \( y = mx + c \), is most frequently utilized due to its straightforwardness.This format offers:

• \( m \), which stands for the slope of the line.
• \( c \), the \( y \text{-intercept} \), demonstrating where the line crosses the y-axis.

Line equations pinpoint the precise direction and positioning of a line in a coordinate plane.
In the original exercise, the equations \( y = x + 1 \) and \( y = 1 - x \) are both sleekly formatted in slope-intercept form. This makes it a breeze to distinguish their properties:

• The slope for the first line is \(1\).
• The slope for the second line is \(-1\).

By identifying these components swiftly, one can analyze line relationships effortlessly. Line equations bridge algebra and geometry, showcasing how algebraic expressions map onto geometric visualization.