Problem 31 - Introductory Algebra for College Students

Question

In Exercises \(31-36,\) determine whether the lines through each pair of points are parallel, perpendicular, or neither. $$(-2,-5) \text { and }(3,10) ;(-1,-9) \text { and }(4,6)$$

Step-by-Step Solution

Answer

The lines passing through the pairs of points are parallel.

Step 1: Calculate the slope of the first pair of points

First we calculate the slope of the line passing through the points (-2, -5) and (3, 10) using the formula \(m = \frac{y2 - y1}{x2 - x1}\). Substitute (-2, -5) for (x1, y1) and (3, 10) for (x2, y2) to get \(m1 = \frac{10 - (-5)}{3 - (-2)} = 3\). So, the slope of the first line is 3.

Step 2: Calculate the slope of the second pair of points

Next, we calculate the slope of the line passing through the points (-1, -9) and (4, 6) using the same formula. Substitute (-1, -9) for (x1, y1) and (4, 6) for (x2, y2) to get \(m2 = \frac{6 - (-9)}{4 - (-1)} = 3\). So, the slope of the second line is also 3.

Step 3: Compare the slopes

Now that we have the slopes of the two lines, we compare them. If the slopes are the same, the lines are parallel. If the product of the slopes is -1, then the lines are perpendicular. Otherwise, the lines are neither parallel nor perpendicular. In this case, both slopes are 3, hence these two lines are parallel.

Key Concepts

Understanding Slope Calculation

To determine if lines are parallel, perpendicular, or neither, we first need to calculate their slopes. The formula for slope (\( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula captures the change in the vertical direction divided by the change in the horizontal direction — effectively showing how "steep" a line is.

Imagine you're on a hike. The slope is like figuring out how steep your path is: a higher slope means a steeper climb. Calculating the slope provides insight into the direction and steepness of a line on the coordinate plane.

For example:

• If the slope is positive, the line ascends as it moves from left to right.
• If the slope is negative, the line descends.
• A slope of zero indicates a horizontal line, while an undefined slope means the line is vertical.

Perpendicular Lines and Their Relationship

Perpendicular lines have a unique relationship when it comes to their slopes. If two lines are perpendicular, their slopes are special in the sense that the product of their slopes is \(-1\). For example, if the slope of one line is 2, then the slope of a line perpendicular to it would be \(-\frac{1}{2}\). This unique relationship arises because perpendicular lines intersect to create a 90-degree angle.

To visualize, if you imagine two paper rectangles intersecting like a cross (one horizontal, one vertical), these edges form perpendicular lines.

Understanding when lines are perpendicular involves:

• Checking for the product of the slopes being \(-1\).
• If one slope is vertical, the perpendicular would be horizontal.

Remember, as soon as you confirm the product is \(-1\), you identify that the lines form a right angle.

Exploring Coordinate Geometry

Coordinate geometry is the study of geometric figures graphed on a plane defined by two perpendicular number lines, the x-axis and y-axis. The plane is effectively a grid where each point is defined by a pair of numbers, known as coordinates. This powerful system allows for analysis of shapes, lines, and angles using algebraic principles. By plotting points and understanding their positions:

• You can calculate distances between points using the distance formula.
• Midpoints between two coordinates can be identified, offering insight into symmetry or center points.
• Angles and directional relationships, like parallel and perpendicular lines, can also be effectively analyzed.

Coordinate geometry merges algebra with geometry to make complex insights about figures and their interactions on a plane both simple and clear.

Through solving problems in coordinate geometry, like determining the nature of lines between points, you employ both graphical and algebraic skills, allowing you to visualize and calculate relationships precisely.