Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2 . Is each pair of lines parallel, perpendicular, or neither? Line 1: Passes through (0,6) and (3,-24) Line 2: Passes through (-1,19) and (8,-71)

Step-by-Step Solution

Answer

The lines are parallel, both having a slope of -10.

Step 1: Identify Points for Line 1

Line 1 passes through the points (0,6) and (3,-24). We will use these points for calculation.

Step 2: Calculate Slope of Line 1

The slope \( m \) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, substituting \((x_1, y_1) = (0, 6)\) and \((x_2, y_2) = (3, -24)\) into the formula: \[ m_1 = \frac{-24 - 6}{3 - 0} = \frac{-30}{3} = -10 \] So, the slope of Line 1 is \(-10\).

Step 3: Identify Points for Line 2

Line 2 passes through the points (-1,19) and (8,-71). We will use these points for calculation.

Step 4: Calculate Slope of Line 2

Using the slope formula for Line 2 with \((x_1, y_1) = (-1, 19)\) and \((x_2, y_2) = (8, -71)\), we have: \[ m_2 = \frac{-71 - 19}{8 - (-1)} = \frac{-90}{9} = -10 \] Thus, the slope of Line 2 is also \(-10\).

Step 5: Compare Slopes to Determine Relationship

Since both slopes \( m_1 \) and \( m_2 \) are equal to \(-10\), the lines are parallel by definition, as parallel lines have equal slopes.

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never meet. They remain the same distance apart over their entire length. This concept is crucial in geometry, especially in shapes like rectangles and squares, where opposite sides are parallel.

To determine if two lines are parallel, we check their slopes:

• If two lines have the same slope, then they are parallel.
• If they have different slopes, they are not parallel.

In the problem provided, both Line 1 and Line 2 have slopes of (-10). Although they pass through different points on the plane, their slopes being identical confirms that the lines are parallel. Understanding this concept helps in recognizing patterns and properties of various geometric shapes.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). An important property of perpendicular lines in coordinate geometry is the relationship between their slopes.

For two lines to be perpendicular:

• Their slopes, when multiplied together, must equal (-1).
• If the slope of one line is \(m\), the slope of the other line should be \(-\frac{1}{m}\).

Unlike parallel lines, perpendicular lines create angles and are often used in architecture and engineering. In our given example, since both slopes are at (-10), the condition \(-10 \times (-10) = -1\) is not satisfied, meaning the lines are not perpendicular. Knowing this relationship helps in the design and analysis of intersecting structures.

Slope Formula

The slope of a line indicates its steepness and direction. The slope formula is key to determining this characteristic of a line. This formula is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1}\]where:

• \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
• \(m\) is the slope.

Using this formula, you can determine whether lines are parallel, perpendicular, or neither. Applying the slope formula involves substituting the coordinates of two points that a line passes through. For instance, in the given exercise, calculating the slopes of Line 1 \((-10)\) and Line 2 \((-10)\) showed us the lines are parallel. Mastery of this formula is essential for solving a wide range of coordinate geometry problems easily.