Problem 50 - 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 (1,7) and (5,5) Line 2 : Passes through (-1,-3) and (1,1)

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Find Slope of Line 1

To find the slope (\(m_1\)) of Line 1, we use the formula for the slope, \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, the points are (1,7) and (5,5). Substituting in the values, we get \( m_1 = \frac{5 - 7}{5 - 1} = \frac{-2}{4} = -\frac{1}{2} \).

Step 2: Find Slope of Line 2

Similarly, to find the slope (\(m_2\)) of Line 2, we use the same formula. For Line 2, the points are (-1,-3) and (1,1). Substituting in the values, we have \( m_2 = \frac{1 - (-3)}{1 - (-1)} = \frac{4}{2} = 2 \).

Step 3: Determine Relationship Between Lines

To determine if the lines are parallel, perpendicular, or neither, we compare their slopes. Parallel lines have equal slopes and perpendicular lines' slopes are negative reciprocals of each other. In this case, \(m_1 = -\frac{1}{2}\) and \(m_2 = 2\). The product of the slopes \( m_1 \times m_2 = -1 \) which means the lines are perpendicular.

Key Concepts

Parallel Lines

Parallel lines are an interesting concept in geometry. They are lines that never meet, no matter how far you extend them in either direction. For two lines to be parallel, they must have the same slope. This means that the steepness of the lines is identical, and they rise over run at the same rate.

To check if two lines are parallel, first calculate the slope of each line using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

After finding the slopes, if they are equal, then the lines are parallel.

In the original exercise, Line 1 and Line 2 had slopes \(-\frac{1}{2}\) and \(2\), respectively. Since these slopes are not equal, the lines are not parallel.

Perpendicular Lines

Perpendicular lines add a fun twist to the concept of line slopes. They intersect at right angles, creating a "T" shape on the plane. For two lines to be perpendicular, the product of their slopes should be \(-1\). This can also be observed when one slope is the negative reciprocal of the other.

Here's how you determine if lines are perpendicular:

• First, find the slopes of both lines using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• Check if the slopes multiply to \(-1\).

In the given problem, the slopes of Line 1 and Line 2 are \(-\frac{1}{2}\) and \(2\). Their product, \(-\frac{1}{2} \times 2\), is indeed \(-1\). Thus, these lines are perpendicular.

Equation of a Line

Understanding the equation of a line is crucial for analyzing the relationship between two lines. One common form used to express a line is the slope-intercept form, which is given by:

• \( y = mx + b \)

Here, "\(m\)" is the slope, and "\(b\)" is the y-intercept where the line crosses the y-axis. Knowing the slope of a line is fundamental when writing its equation.

For example, if a line passes through the point (1,7) with a slope of \(-\frac{1}{2}\), you can substitute these values into the slope-intercept form to find the equation:

• Substitute \(m = -\frac{1}{2}\) and point (1,7) into \( y = mx + b \) to get \(7 = -\frac{1}{2}(1) + b \)
• Solve for \(b\)

This results in the line equation \( y = -\frac{1}{2}x + 7.5 \).

It's always essential to start with the foundational formulas and build upon them with given points to derive other parameters of the line like its equation.