Problem 47 - College Algebra

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.

Step 1: Calculate the Slope of Line 1

To find the slope of Line 1, use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, the points are \((0, 6)\) and \((3, -24)\). Substitute these into the formula: \( m_1 = \frac{-24 - 6}{3 - 0} = \frac{-30}{3} = -10 \).

Step 2: Calculate the Slope of Line 2

Similarly, use the slope formula for Line 2 with the points \((-1, 19)\) and \((8, -71)\): \( m_2 = \frac{-71 - 19}{8 - (-1)} = \frac{-90}{9} = -10 \).

Step 3: Determine if Lines are Parallel, Perpendicular, or Neither

Since the slopes of Line 1 and Line 2 are both \(-10\), the lines are parallel. Two lines are parallel if they have the same slope.

Key Concepts

Parallel Lines

When discussing parallel lines, we refer to lines that run side by side and never intersect. A key characteristic of parallel lines is that they have the same slope. In simple terms, the slope defines the steepness or the angle of the line. So, when two lines are parallel, they rise or fall at the same angle.
For example, if Line 1 has a slope of \(-10\) and Line 2 also has a slope of \(-10\), this means both lines are moving in the same direction at the same rate. No matter how far you extend these lines on a graph, they will never cross each other.

• Identifying Parallel Lines: Just compare the slopes of two lines. If both slopes are equal, the lines are parallel.
• Example in Coordinate Plane: In our exercise, the slopes \(m_1 = -10\) and \(m_2 = -10\) confirm that both lines are parallel.

Perpendicular Lines

Perpendicular lines are fascinating because they intersect each other at a right angle, forming a perfect 90-degree angle. This is very different from parallel lines. The special property of perpendicular lines is in their slopes: if the slope of one line is \(m\), the slope of the line perpendicular to it will be the negative reciprocal, which is \(-1/m\).

• Understanding Negative Reciprocal: If the slope of one line is \(-10\), a line perpendicular to it would have a slope of \(rac{1}{10}\) because \(-1/-10 = rac{1}{10}\).
• Checking Perpendicularity: If you multiply the slopes of two lines and get \(-1\), these lines are perpendicular.

In our exercise, the lines are parallel since both slopes are \(-10\) rather than being negative reciprocals of each other.

Slope Formula

The slope formula is a straightforward mathematical tool used to calculate the incline of a line. It helps tell us how much a line goes up or down as it moves from left to right. The formula is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• Breaking Down the Formula: \(y_2\) and \(y_1\) are the y-coordinates of any two points on a line, while \(x_2\) and \(x_1\) are the x-coordinates. Subtracting these values gives us the difference in the \(y\)-values over the difference in the \(x\)-values.
• Application: Understanding the formula allows you to calculate the slope between any two points. In our exercise, applying it to both Line 1 and Line 2 helped us find that both lines had the same slope of \(-10\).

Knowing how to use this formula is crucial when working with equations of lines and understanding how lines behave on a graph.