Problem 59 - Algebra for College Students

Question

Determine whether the lines \(l_{1}\) and \(l_{2}\) are parallel, perpendicular, or neither. \(l_{1}\) goes through \((-2,-3)\) and \((4,1), l_{2}\) goes through \((-1,7)\) and \((1,4)\)

Step-by-Step Solution

Answer

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

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

The formula to calculate the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For line \(l_{1}\), the points are \((-2, -3)\) and \((4, 1)\). So, the slope of \(l_{1}\) is \(m_{1} = \frac{1 - (-3)}{4 - (-2)} = \frac{4}{6} = \frac{2}{3}\).

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

Using the same formula, calculate the slope for line \(l_{2}\) given the points \((-1, 7)\) and \((1, 4)\). The slope of \(l_{2}\) is \(m_{2} = \frac{4 - 7}{1 - (-1)} = \frac{-3}{2}\).

Step 3: Compare the slopes of \(l_{1}\) and \(l_{2}\)

Lines are parallel if their slopes are equal, and they are perpendicular if the product of their slopes is \(-1\). For \(l_{1}\) and \(l_{2}\), the slopes are \(\frac{2}{3}\) and \(-\frac{3}{2}\) respectively. The product of the slopes is \( \left( \frac{2}{3} \right) \times \left( -\frac{3}{2} \right) = -1\). Because the product of the slopes is \(-1\), the lines \(l_{1}\) and \(l_{2}\) are perpendicular.

Key Concepts

Slope Calculation

To determine if lines are parallel, perpendicular, or neither, we first need to calculate their slopes. The slope represents how steep a line is. For any line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] The slope calculation involves subtracting the \(y\) coordinates and dividing by the difference in \(x\) coordinates. Keep in mind:

• If the slope is positive, the line inclines upwards.
• If the slope is negative, the line declines downwards.
• A zero slope means the line is horizontal.
• An undefined slope (division by zero) means the line is vertical.

In our example, the slopes of line \( l_1 \) and line \( l_2 \) are calculated as follows:

First, for \( l_1 \) through points \((-2,-3)\) and \((4,1)\):
\( m_1 = \frac{1 - (-3)}{4 - (-2)} = \frac{4}{6} = \frac{2}{3} \)

Next, for \( l_2 \) through points \((-1,7)\) and \((1,4)\):
\( m_2 = \frac{4 - 7}{1 - (-1)} = \frac{-3}{2} \)

Lines and Angles

When comparing lines, their slopes give us insights into their relationship. Specifically, slopes help us identify if lines are parallel, perpendicular, or neither. Parallel lines have the same slope. They never intersect and are always equidistant from each other. In the context of slopes:

• If \(m_1 = m_2\), the lines are parallel.

Perpendicular lines intersect at right angles (\(90^\text{°}\)). The slopes of perpendicular lines have a unique property—their slopes are negative reciprocals of each other:

• If \(m_1 \cdot m_2 = -1\), the lines are perpendicular.

In the given problem, after calculating the slopes, we find:
- Slope of \( l_1 \) is \( \frac{2}{3} \)
- Slope of \( l_2 \) is \( -\frac{3}{2} \) The product of these slopes is \[ \left( \frac{2}{3} \right) \times \left( -\frac{3}{2} \right) = -1 \] Therefore, lines \( l_1 \) and \( l_2 \) are perpendicular.

Coordinate Geometry

Coordinate geometry helps us represent and analyze geometric figures using coordinates on a plane. This branch is essential for understanding the positions and relationships between various geometric entities like points, lines, and shapes. In the coordinate plane:

• A point is represented as an ordered pair \((x, y)\).
• A line can be defined by two points.
• We can use formulas to calculate distance, midpoint, and slopes.

For our exercise, consider points \((-2, -3)\), \((4, 1)\), \((-1, 7)\), and \((1, 4)\). These represent locations on the plane through which lines \( l_1 \) and \( l_2 \) pass. By plotting these points and using the slope formula, we can understand the orientation and relationship of these lines visually and mathematically. Therefore, coordinate geometry not only involves calculations but also provides a visual understanding of geometric concepts and their relationships.