Problem 55 - Algebra for College Students

Question

Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. In goes through \((0,2)\) and \((7,9), l_{2}\) goes through \((0,-3)\) and \((1,-2)\)

Step-by-Step Solution

Answer

The lines are parallel.

Step 1 - Calculate the slope of line \( l_1 \)

The slope of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the formula: \m = \frac{y_2 - y_1}{x_2 - x_1}\. For line \( l_1 \) passing through \( (0, 2) \) and \( (7, 9) \), the slope is \m_1 = \frac{9 - 2}{7 - 0} = \frac{7}{7} = 1\.

Step 2 - Calculate the slope of line \( l_2 \)

Using the same formula, for line \( l_2 \) passing through \( (0, -3) \) and \( (1, -2) \), the slope is \m_2 = \frac{-2 - (-3)}{1 - 0} = \frac{1}{1} = 1\.

Step 3 - Compare the slopes

Since both slopes \( m_1 \) and \( m_2 \) are equal to 1, the lines are parallel. Lines are parallel if their slopes are equal.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is crucial in coordinate geometry. The slope of a line is a measure of its steepness and direction. Mathematically, the slope is defined as the ratio of the vertical change (rise) to the horizontal change (run). For two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope \(m\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula essentially tells you how much the y-coordinate (vertical position) changes for a unit change in the x-coordinate (horizontal position). Let's apply this formula to the example provided in the exercise. For line \(l_1\), which goes through points \((0,2)\) and \((7,9)\), we calculate the slope as follows: \[ m_1 = \frac{9 - 2}{7 - 0} = \frac{7}{7} = 1 \] Similarly, for line \(l_2\), which goes through points \((0,-3)\) and \((1,-2)\), the slope is: \[ m_2 = \frac{-2 - (-3)}{1 - 0} = \frac{1}{1} = 1 \] The calculated slopes are used to compare line characteristics such as being parallel or perpendicular.

Parallel Lines

Parallel lines are an important concept in coordinate geometry. Two lines are said to be parallel if they have the same slope but are not coincident (they do not intersect). This means that the lines run in the same direction and are spaced equally apart. To determine if two lines are parallel, we compare their slopes: \[\text{If } m_1 = m_2, \text{ then the lines are parallel.}\] In the given exercise, the slopes of both lines \(l_1\) and \(l_2\) were found to be equal to 1. Hence, \(l_1\) is parallel to \(l_2\). Understanding parallel lines helps in various fields such as engineering, architecture, and graphics, where maintaining parallelism ensures stability and aesthetics.

Coordinate Geometry

Coordinate Geometry, also known as analytic geometry, involves the study of geometric figures through a coordinate system. By representing geometric shapes and lines using algebraic equations, we can analyze and solve geometric problems more efficiently. The basic elements in coordinate geometry are points, lines, and curves. Points are represented by their coordinates \((x, y)\) in a plane. Lines can be described by equations like y = mx + b, where \(m\) is the slope and \(b\) is the y-intercept. Analyzing the relationships between different lines, such as determining parallelism or perpendicularity, is a fundamental operation. In the given exercise, coordinate geometry principles help identify that lines \(l_1\) and \(l_2\) are parallel by comparing their slopes. This analytical approach allows for precise calculations and visual representations, enhancing our understanding of spatial relationships and geometrical properties.