Problem 51 - College Algebra

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

Step-by-Step Solution

Answer

The lines are parallel since they have the same slope of -2.

Step 1: Identifying the Points

Line 1 passes through the points (2,5) and (5,-1). Line 2 passes through the points (-3,7) and (3,-5). We need these to find the slopes of both lines.

Step 2: Calculating the Slope of Line 1

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For Line 1, \((x_1, y_1) = (2, 5)\) and \((x_2, y_2) = (5, -1)\). Thus, the slope \( m_1 = \frac{-1 - 5}{5 - 2} = \frac{-6}{3} = -2 \).

Step 3: Calculating the Slope of Line 2

Using the same slope formula, for Line 2 with points \((x_1, y_1) = (-3, 7)\) and \((x_2, y_2) = (3, -5)\), we find the slope \( m_2 = \frac{-5 - 7}{3 - (-3)} = \frac{-12}{6} = -2 \).

Step 4: Comparing the Slopes

Two lines are parallel if they have the same slope and are perpendicular if the product of their slopes is -1. Here, the slopes of Line 1 and Line 2 are both -2. Therefore, both lines are parallel.

Key Concepts

Slope Calculation

Calculating the slope of a line is an essential skill in geometry. The slope provides a numerical value that represents the steepness and direction of a line.
A line's slope is calculated using the coordinates of two distinct points on the line. Here's the formula for slope calculation:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• The symbol \( m \) represents the slope.
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• Subtract the y-coordinates and the x-coordinates separately to use in the formula.

Using these simple subtractions, we can determine the slope of Line 1 to be -2 and Line 2 also to be -2.
Intuitively, the slope tells us how much the line rises or falls as it moves from left to right. A negative slope means the line is descending.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to determine geometric relationships using algebra. It lets us use numerical equations to describe lines, curves, and shapes.
By assigning coordinates to points, the position and connection of these points are mathematically described. The coordinate system defines positions using ordered pairs, like (2,5) or (-3,7).

• On a coordinate plane, the horizontal line is called the x-axis and the vertical line is the y-axis.
• Coordinates are noted as \((x, y)\), describing a point's location relative to these axes.

In this context, we used the coordinates of two points on each line to compute the slope, demonstrating how geometry and algebra work together to solve spatial problems.

Line Equations

Line equations are foundational in coordinate geometry for defining straight lines. With a calculated slope and a point through which the line passes, we can write its equation in the slope-intercept form:\[y = mx + b\]

• \( m \), the slope, is known from our calculations.
• \( b \), the y-intercept, can be found by substituting one point's coordinates into the equation.

For example, with the slope of -2 for both lines, you can substitute \( m = -2 \) into the equation to find \( b \) using any point on the line.
Understanding line equations allows you to see how lines are expressed algebraically and how they behave on a graph. This knowledge helps determine relationships between lines, such as whether they are parallel or intersecting.