Problem 56 - Algebra for College Students

Question

Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((4,3)\) and \((2,6), l_{2}\) goes through \((0,0)\) and \((3,2)\)

Step-by-Step Solution

Answer

The lines \(l_{1}\) and \(l_{2}\) are perpendicular.

Step 1: Calculate the slope of line \(l_{1}\)

The slope of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For line \(l_{1}\) going through points \((4,3)\) and \((2,6)\), calculate the slope: \[ m_{1} = \frac{6 - 3}{2 - 4} = \frac{3}{-2} = -\frac{3}{2} \] Therefore, the slope of line \(l_{1}\) is -\(\frac{3}{2}\).

Step 2: Calculate the slope of line \(l_{2}\)

Using the same formula for the slope, calculate the slope of line \(l_{2}\) passing through points \((0,0)\) and \((3,2)\): \[ m_{2} = \frac{2 - 0}{3 - 0} = \frac{2}{3} \] Therefore, the slope of line \(l_{2}\) is \(\frac{2}{3}\).

Step 3: Compare the slopes

To determine if lines are parallel, perpendicular, or neither, compare their slopes: - Parallel lines have identical slopes. - Perpendicular lines have slopes that are negative reciprocals. - Otherwise, lines are neither. Compare the slopes of \(l_{1}\) and \(l_{2}\): - Slope of \(l_{1}\): -\(\frac{3}{2}\) - Slope of \(l_{2}\): \(\frac{2}{3}\). The product of the slopes is: \[ -\frac{3}{2} \times \frac{2}{3} = -1 \] Since the product is -1, the lines are perpendicular.

Key Concepts

slope calculation

To understand the relationship between two lines, we need to start by calculating their slopes. The slope of a line indicates its steepness and direction. It is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula requires two points on the line, \(x_1, y_1\) and \(x_2, y_2\).For our example, we first evaluate the slope of line \(l_1\), which passes through points \(4, 3\) and \(2, 6\). Plugging these coordinates into the formula, we find:\[ m_1 = \frac{6 - 3}{2 - 4} = \frac{3}{-2} = -\frac{3}{2} \]Thus, the slope of \(l_1\) is negative three-halves or -1.5.Next, we calculate the slope of line \(l_2\), which passes through points \(0, 0\) and \(3, 2\). Using the same formula, we get:\[ m_2 = \frac{2 - 0}{3 - 0} = \frac{2}{3} \]So, the slope of \(l_2\) is two-thirds or approximately 0.67.
Understanding these slopes is crucial as they form the basis for determining the relationship between the lines.

coordinate geometry

Coordinate geometry helps us precisely locate points on a plane using ordered pairs such as \(x, y\). This approach simplifies complex geometry problems by converting them into algebraic problems. In our example, when you're given points \[ (4, 3) \] and \[ (2, 6) \] for line \(l_1\), the slopes are quickly found using their coordinates. Similarly, \[ (0, 0) \] and \[ (3, 2) \] are used to find the slope of \(l_2\).
We closely examine these coordinate points to understand the nature of the lines. Once the slopes are calculated, we can plot these points on a graph to visualize the lines’ inclinations. The process reinforces our understanding of how algebra and geometry interlink.
Coordinate geometry is especially useful because it allows us to move seamlessly from visual representations to exact numeric analysis, enhancing our ability to solve problems effectively.

line comparison

Comparing lines lets us determine whether they are parallel, perpendicular, or neither. Here's a quick guide:

• Parallel lines have identical slopes \(m_1 = m_2\). They never intersect and are always at the same distance apart.
• Perpendicular lines have slopes that are negative reciprocals, meaning \(m_1 \times m_2 = -1\). They intersect at a right angle (90 degrees).
• If the slopes do not fit the above criteria, the lines are neither parallel nor perpendicular.

For our lines \(l_1\) and \(l_2\), the slopes are \(-\frac{3}{2} \) and \(\frac{2}{3}\) respectively. The product of these slopes is calculated as follows:\[ -\frac{3}{2} \times \frac{2}{3} = -1 \]Since the product is -1, \(l_1\) and \(l_2\) are perpendicular. Recognizing these relationships helps in various applications, including geometry problems, computer graphics, and even in real-world construction projects where understanding angles and intersections is vital.
Developing this skill is foundational for more advanced mathematical concepts.