Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. \(\begin{array}{ll}(-1,3) & \text { and }(5,1) \\ (-2,3) & \text { and }(0,9)\end{array}\)

Step-by-Step Solution

Answer

The lines are perpendicular.

Step 1: Find the Slope Formula

The formula to find the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). We will apply this formula to both pairs of points.

Step 2: Calculate the Slope for the First Pair

For the points \((-1,3)\) and \((5,1)\), use the formula: \[ m = \frac{1 - 3}{5 - (-1)} \]This simplifies to:\[ m = \frac{-2}{6} = -\frac{1}{3} \]Thus, the slope of the line through these points is \(-\frac{1}{3}\).

Step 3: Calculate the Slope for the Second Pair

For the points \((-2,3)\) and \((0,9)\), use the formula: \[ m = \frac{9 - 3}{0 - (-2)} \]This simplifies to:\[ m = \frac{6}{2} = 3 \]Thus, the slope of the line through these points is \(3\).

Step 4: Determine If Lines Are Parallel or Perpendicular

Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is \(-1\). Here we have slopes \(-\frac{1}{3}\) and \(3\). Multiplying these gives:\[ -\frac{1}{3} \times 3 = -1 \]This indicates the lines are perpendicular.

Key Concepts

Parallel Lines

Parallel lines are an important concept in geometry and are often found in various mathematical problems. Two distinct, non-vertical lines are said to be parallel if and only if they have the same slope. This means that no matter how far the lines extend, they will never intersect. Parallel lines share the same steepness and direction.

• For example, if one line has a slope of 2, then a line parallel to it will also have a slope of 2.
• Parallel lines are commonly denoted as \( L_1 \parallel L_2 \).

Parallel lines appear visually similar but may be at different vertical positions due to different y-intercepts.
Understanding parallel lines is crucial for identifying features of shapes and solving systems of equations. Recognizing them quickly helps streamline solving mathematical equations and proofs.

Perpendicular Lines

Perpendicular lines intersect to form a right angle, which is precisely 90 degrees. The defining feature of perpendicular lines in the coordinate plane is their slope product. If the product of their slopes equals

• -1, then the lines are perpendicular.

For example, consider two lines where one line has a slope of \( m_1 = \frac{1}{2} \) and the other line has a slope of \( m_2 = -2 \) (since the negative reciprocal of \( \frac{1}{2} \) is \(-2\)), their slope product is \( \frac{1}{2} \times -2 = -1 \).
In the given problem, we found two slopes: \(-\frac{1}{3}\) and \(3\). Calculating their product yields \(-1\), establishing that these lines are indeed perpendicular. This characteristic makes perpendicular lines especially useful in coordinate geometry, allowing us to quickly determine special relationships in shapes and graphs.

Slope Formula

The slope formula is a fundamental tool in coordinate geometry, employed to determine the steepness or incline of a line between two points. Given two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

• This formula measures the vertical change (rise) against the horizontal change (run).
• A positive slope indicates an upward direction from left to right, while a negative slope indicates a downward direction.
• A zero slope implies a horizontal line, and an undefined slope signifies a vertical line.

In the given problem, to find the slopes, we applied this formula to each pair of points. Calculating accurately establishes if lines are parallel, perpendicular, or neither by comparing their slopes. Using the slope formula is a key skill when analyzing line graphs and solving problems in algebra and calculus.