Problem 51 - 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 (2,5) and (5,-1) Line 2: Passes through (-3,7) and (3,-5)

Step-by-Step Solution

Answer

Both lines are parallel because their slopes are equal.

Step 1: Calculate the Slope of Line 1

To find the slope of Line 1, use the formula for slope: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points (2,5) and (5,-1), we find \( m = \frac{-1 - 5}{5 - 2} = \frac{-6}{3} = -2 \). So, the slope of Line 1 is -2.

Step 2: Calculate the Slope of Line 2

To find the slope of Line 2, use the same formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points (-3,7) and (3,-5), we find \( m = \frac{-5 - 7}{3 + 3} = \frac{-12}{6} = -2 \). So, the slope of Line 2 is also -2.

Step 3: Determine the Relationship Between the Lines

Lines are parallel if they have equal slopes and perpendicular if the product of their slopes is -1. Since both slopes are -2, Line 1 and Line 2 are parallel.

Key Concepts

Parallel Lines

Parallel lines are fascinating because they never cross each other; they extend infinitely in the same direction without intersecting. The key characteristic of parallel lines is their slopes. When two lines are parallel, they have the same slope. It means they rise and run at the same angle. Think of train tracks or power lines that run alongside each other forever.
In our exercise, we calculated the slopes of both Line 1 and Line 2: both were found to be -2. This matching slope is a tell-tale confirmation that the lines are parallel.
When solving problems involving parallel lines, remember these points:

• Check if slopes are equal.
• If lines have the same slope, they will never intersect.
• Even if their intercepts differ, they remain parallel.

Perpendicular Lines

Perpendicular lines have a special relationship where they intersect at a right angle (90 degrees). This is in stark contrast to parallel lines. The magic number for perpendicular lines is -1, but why? It's all in the slopes.
If you multiply the slopes of two perpendicular lines, you get -1. Let's delve deeper:
Suppose Line A has a slope of 3. For a line to be perpendicular to it, its slope must be -1/3. Note how the slopes are negative reciprocals of each other.

• Perpendicular lines intersect at right angles.
• The product of their slopes is always -1.
• If one line's slope is 'm', the other must be '-1/m'.

Understanding this concept adds a strong tool to your math toolkit!

Linear Equations

Linear equations form the backbone of much of algebra, and understanding them is crucial for grasping more advanced mathematical concepts. A linear equation generally takes the form of \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. This equation describes a straight line in a coordinate plane.
Let's break it down:

• Slope \( m \): Tells us how steep the line is.
• Y-intercept \( b \): The point where the line crosses the y-axis.

Linear equations are great because they are easy to graph and solve. You just need two points to sketch a line, and you can find those by choosing any two x-values, substituting them into the equation to find corresponding y-values.
In real-life problems, linear equations can model relationships between variables that change at constant rates.