Problem 46 - 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. $$\begin{array}{l} (2,5) \text { and }(5,9) \\ (-1,-1) \text { and }(2,3) \end{array}$$

Step-by-Step Solution

Answer

The lines are parallel, both have a slope of \(\frac{4}{3}\).

Step 1: Recall the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This formula will be used to calculate the slope for each pair of points provided.

Step 2: Calculate the Slope of the First Line

Use the points \((2, 5)\) and \((5, 9)\). Plug these into the formula: \(m_1 = \frac{9 - 5}{5 - 2} = \frac{4}{3}\). So, the slope of the first line is \(\frac{4}{3}\).

Step 3: Calculate the Slope of the Second Line

Use the points \((-1, -1)\) and \((2, 3)\). Plug these into the formula: \(m_2 = \frac{3 - (-1)}{2 - (-1)} = \frac{4}{3}\). So, the slope of the second line is \(\frac{4}{3}\).

Step 4: Determine if the Lines are Parallel or Perpendicular

Lines are parallel if they have the same slope. Since \(m_1 = \frac{4}{3}\) and \(m_2 = \frac{4}{3}\), the lines are parallel. Lines are perpendicular if the product of their slopes is \(-1\). Here, the product is \(\frac{4}{3} \times \frac{4}{3} = \frac{16}{9}\), which is not \(-1\). Thus, the lines are not perpendicular.

Key Concepts

Parallel Lines

Parallel lines are fascinating in geometry and everyday life. They consist of two or more lines that never meet, regardless of how far they extend in either direction. This unique property hinges on their slopes. In mathematics, if the slopes of two lines are equal, the lines are parallel.

When we talk about slopes, it helps to picture this concept. Imagine lines that are laid along train tracks, each track running at the same angle forever. These tracks (or lines) never cross each other, making them parallel.

• If Line 1 has a slope of \(m_1\) and Line 2 has a slope of \(m_2\), the relationship for parallelism is \(m_1 = m_2\).
• This is why the lines through the points \((2,5)\) and \((5,9)\), and the points \((-1,-1)\) and \((2,3)\) are parallel; both have a slope of \(\frac{4}{3}\).

A great way to test for parallel lines is by comparing equations where the coefficients of \(x\) (which represent the slope) should be the same once simplified down.

Perpendicular Lines

Perpendicular lines bring a different perspective to geometry. These lines intersect, and they do so at right angles, forming 90 degrees at their junction.

Identifying perpendicular lines also involves slopes. For two lines to be perpendicular, the product of their slopes must be exactly -1.

• Mathematically, this condition is expressed as \(m_1 \times m_2 = -1\).
• If Line 1 has a slope of \(\frac{4}{3}\) and Line 2 also has \(\frac{4}{3}\), then the product is \(\frac{16}{9}\), not -1, revealing they aren't perpendicular.

Visualizing this can be fun! Imagine a plus sign or the intersecting beams of a window pane forming right angles. Athletes running on tracks might intersect without crashing, thanks to perpendicular guidelines keeping them on course. This concept is crucial in fields like architecture, engineering, and art design for precise, intersecting lines at exact angles.

Slope Formula

Understanding the slope formula is fundamental in connecting different points on a coordinate plane. The slope is essentially a measure of the steepness of a line.

To calculate a 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 coordinates of two points the line passes through.
• The formula calculates how much \(y\) changes per unit change in \(x\).

Picture hiking up a mountain: the steeper it is, the greater the slope. The weather graph on news channels is another excellent application, showing temperature changes over time with varying slopes.

In our specific exercise example, both line segments have points that yield a rise of 4 and a run of 3, which simplifies the slope to \(\frac{4}{3}\). Recognizing and calculating slope is a powerful tool, useful in physics, data analysis, navigation, and even everyday decision making.