Problem 33 - Introductory Algebra for College Students

Question

In Exercises \(27-30\), determine whether the lines through each pair of points are perpendicular. $$(-4,-12) \text { and }(0,-4) ;(0,-5) \text { and }(2,-4)$$

Step-by-Step Solution

Answer

No, the lines are not perpendicular because the product of their slopes is not equal to -1.

Step 1: Find the slope of the first line

The slope of the first line is given by the formula \((y_2 - y_1) / (x_2 - x_1)\). We use the first two points \( (-4, -12) \) and \((0, -4)\) to obtain the slope. So here, \( x_1 = -4, y_1 = -12, x_2 = 0, \) and \( y_2 = -4\). We substitute these into the formula to get \( (-4 - (-12)) / (0 - (-4)) = 2\).

Step 2: Find the slope of the second line

We do the same for the second line. We have the points (0, -5) and (2, -4). So, we have \( x_1 = 0, y_1 = -5, x_2 = 2, \) and \( y_2 = -4\). Substituting these into the formula gives \( (-4 - (-5)) / (2 - 0) = 0.5\).

Step 3: Check if the lines are perpendicular

Lines are perpendicular if the product of their slopes is -1, therefore we multiply the two slopes: \(2 * 0.5 = 1 \). Since 1 is not equal to -1, we can say that the lines are not perpendicular.

Key Concepts

Slope Calculation

Calculating the slope of a line is a fundamental concept in coordinate geometry. To find the slope between two points, we use the formula \((y_2 - y_1) / (x_2 - x_1)\). Here,

• \(x_1\) and \(y_1\) are the coordinates of the first point,
• \(x_2\) and \(y_2\) are the coordinates of the second point.

This calculation gives us an idea of how steep the line is. A positive slope means the line rises from left to right, while a negative slope means it falls.
In our exercise, we calculated the slope for two lines. The procedure involved selecting appropriate points and substituting their values into the slope formula. For the first line, using points \((-4, -12)\) and \((0, -4)\), we calculated the slope as 2. For the second line with points \((0, -5)\) and \((2, -4)\), the slope came out to be 0.5.

Linear Equations

Linear equations are equations of the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This is called the slope-intercept form of a line. These equations describe straight lines in a coordinate system. The slope \(m\) tells us how steep the line is, while the intercept \(b\) indicates where the line crosses the y-axis.Understanding linear equations is crucial when working with lines on a graph. Each point on a line satisfies the linear equation representing that line. This equation reveals the mathematical relationship between the two variables represented on the axes. In our example, from the given points and slopes, we could create such equations.
For instance, the line with a slope of 2 could be expressed as \(y = 2x + b\) after determining what \(b\) is based on points \((-4, -12)\) or \((0, -4)\). Similarly, for the second line, \(y = 0.5x + b\) could be developed.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to describe the spatial relationships between points, lines, and figures in a coordinate system. By using coordinates, we can determine different properties like distance between points, equations of lines, and conditions for perpendicularity or parallelism. When considering lines, they can be examined using their slopes to check relationships like being perpendicular or parallel. Two lines are perpendicular if the product of their slopes is -1.
In our exercise, we confirmed that the lines defined by the given points are not perpendicular because their slope product was 1, not -1. This illustrates how coordinate geometry helps us efficiently determine these relationships using algebraic calculations rather than graphical estimations. It allows a deeper understanding of how geometric properties and algebraic techniques relate.