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

Step-by-Step Solution

Answer

Line 1 and Line 2 are perpendicular.

Step 1: Calculate the Slope of Line 1

The formula for finding the slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, the points are (1, 7) and (5, 5). Substitute these into the formula: \( m = \frac{5 - 7}{5 - 1} = \frac{-2}{4} = -\frac{1}{2} \). Thus, the slope of Line 1 is \(-\frac{1}{2}\).

Step 2: Calculate the Slope of Line 2

Using the same formula for slope as in Step 1, we apply it to Line 2, which passes through the points (-1, -3) and (1, 1). Substitute the values into the formula: \( m = \frac{1 - (-3)}{1 - (-1)} = \frac{1 + 3}{1 + 1} = \frac{4}{2} = 2 \). Thus, the slope of Line 2 is \(2\).

Step 3: Compare Slopes for Parallelism or Perpendicularity

Two lines are parallel if their slopes are equal and perpendicular if the product of their slopes is \(-1\). The slope of Line 1 is \(-\frac{1}{2}\) and the slope of Line 2 is \(2\). Calculate their product: \(-\frac{1}{2} \times 2 = -1\). Since the product equals \(-1\), Line 1 and Line 2 are perpendicular.

Key Concepts

Slope of a Line

The slope of a line is a key concept in algebra and coordinate geometry. It measures the steepness or incline of the line and is usually represented by the letter \( m \). The slope is calculated as the ratio of the vertical change to the horizontal change between two points on the line. This is often referred to as the "rise over run".

To determine the slope, use the formula:

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

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

In practice, calculating the slope can help us understand how quickly values are changing relative to each other. For example, if a line passes through the points \((1,7)\) and \((5,5)\), we would substitute these coordinates into the slope formula to find \( m = -\frac{1}{2} \).

This negative slope indicates that as you move along the line from left to right, the line goes downward. A positive slope would indicate that the line rises as you move from left to right, and a slope of zero would suggest a horizontal line.

Parallel and Perpendicular Lines

Understanding parallel and perpendicular lines is vital in coordinate geometry.

**Parallel Lines**:

• Two lines are parallel if they have the same slope.
• They will never intersect, no matter how far they are extended.
• For example, lines with slopes \( m_1 = 3 \) and \( m_2 = 3 \) are parallel because they share the same slope.

**Perpendicular Lines**:

• Two lines are perpendicular if the product of their slopes is \(-1\).
• This means they intersect at a right angle (90 degrees).
• For example, if Line 1 has a slope of \(-\frac{1}{2}\), and Line 2 has a slope of \(2\), they are perpendicular because \(-\frac{1}{2} \times 2 = -1\).

Grasping these concepts helps in visualizing the arrangement of lines in various geometric shapes and diagrams. When lines are parallel or perpendicular, they have specific geometric properties that can be useful for solving various algebraic and geometric problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a link between algebra and geometry through graphs and equations.

This branch of math uses a coordinate plane where each point is uniquely identified by a pair of numerical coordinates. These coordinates are systematically expressed as \((x, y)\), representing horizontal and vertical distances from a reference point called the origin \((0,0)\).

Coordinate geometry is particularly powerful because it offers a way to describe geometric figures in numerical terms. For instance, the equation of a line, the determination of its slope, and the evaluation of angles between lines are all concepts rooted in coordinate geometry.

• By plotting points like \((1,7)\), \((5,5)\), \((-1,-3)\), and \((1,1)\), we can draw lines and calculate properties like their slopes and relationships.
• It's also useful in real-world applications where one can determine distances and angles just by knowing the coordinates of points.
• Moreover, coordinate geometry allows the derivation and use of formulas like the distance formula and the midpoint formula, which help in analyzing and solving various geometric problems numerically.

These tools and concepts form a fundamental part of algebra and are widely applied in sciences, engineering, and technology.