Problem 27 - Introductory Algebra for College Students

Question

In Exercises \(27-30\), determine whether the lines through each pair of points are perpendicular. \((1,5)\) and \((0,3) ;(-2,8)\) and \((2,6)\)

Step-by-Step Solution

Answer

Yes, the lines are perpendicular.

Step 1: Calculation of First Slope

The formula to calculate the slope, given two points \((x_1, y_1)\) and \((x_2, y_2)\), is given by \[m_1 = (y_2 - y_1)/(x_2 - x_1)\]. Use this formula to calculate the slope for the pair of points \((1,5)\) and \((0,3)\). Plug in the values into the formula: \[m_1 = (3 - 5) / (0 - 1) = -2/-1 = 2.\]

Step 2: Calculation of Second Slope

Now calculate the slope for the second pair of points \((-2,8)\) and \((2,6)\). Using the same formula: \[m_2 = (6 - 8) / (2 - (-2)) = -2/4 = -0.5.\]

Step 3: Determination of Perpendicularity

Now, we need to test if \[m_1 \cdot m_2 = -1\]. Bottling in the values: \[2 \cdot -0.5 = -1\]. Therefore, the lines are perpendicular.

Key Concepts

Algebra

Algebra, the wonderful world where numbers meet letters! It is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols often represent quantities without fixed values, known as variables. In our exercise, algebra sets the stage for finding the slopes of the lines through pairs of points.
Using algebraic methods, particularly equations, we can solve real-world problems by representing them in simplified forms using variables and constants. For example, when calculating slopes, an algebraic formula comes to our rescue:

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

In this linear equation formula, \(x_1, y_1\) are the coordinates of the first point and \(x_2, y_2\) are those of the second. If algebra can transform random data into clear, comprehensible insights, it can do anything!

Slope Calculation

Calculating the slope is like finding the steepness of a hill. You use a simple formula to determine how tilted a line that connects two points is. This is key in understanding many geometric concepts.
For any two points, the slope measures how much the "output" or "height" changes over a specific "input" or "run." Our algebraic formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) comes in handy here.
Simply put:

• Pick two points on the line and label them as \((x_1, y_1)\) and \((x_2, y_2)\).
• Subtract the y-values: \(y_2 - y_1\).
• Subtract the x-values: \(x_2 - x_1\).
• Divide the difference in y-values by the difference in x-values to get your slope.

In our task, we calculated two slopes: one for each line, which helped us determine if the two lines are perpendicular.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, allows us to describe geometric shapes numerically. This system combines algebra and geometry using coordinates (x, y) to denote points on a plane.
The exercise involves understanding these points on a plane and the relationships between lines that connect them. Coordinate geometry makes it easy to calculate distances, angles, and slopes. Imagine you are plotting points on a grid.

• The point \((1, 5)\) pinpoints exactly where it belongs on your grid.
• The line connecting two points forms a visual representation of a relationship between them.
• Slope becomes a measure of how these lines tilt and diverge from each other.

By using coordinate geometry, it's straightforward to verify if two lines have a special angular relationship, like being perpendicular. Here, the task was to show that the product of the slopes equals \(-1\), confirming perpendicularity.