Problem 72 - College Algebra and Calculus: An Applied Approach

Question

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: x-5 y=-2 ; L_{2}:-3 x+15 y=6\)

Step-by-Step Solution

Answer

The lines \(L_{1}\) and \(L_{2}\) are parallel.

Step 1: Convert Both Equations to Slope-Intercept Form (y=mx+c)

First, reshuffle both equations to find the slope value of each line. For \(L_{1}\), the equation becomes \(y = (1/5)x + (2/5)\) after rearranging. Similarly for \(L_{2}\), the equation becomes \(y = (1/5)x - (2/5)\).

Step 2: Compare Slopes of Both Lines

The slope \(m\) of \(L_{1}\) is 1/5 and of \(L_{2}\) is also 1/5. As the slopes of the two lines are the same, the lines are parallel.

Step 3: Check if Slopes are Negative Reciprocals

If slopes are negative reciprocals, then lines would be perpendicular. However, in this case, this does not apply as the slopes are equal and not negative reciprocals of each other.

Key Concepts

Slope-Intercept Form

Understanding the "slope-intercept form" is crucial for analyzing and graphing linear equations. It makes equations easier to read and work with. The slope-intercept form is given by the formula \(y = mx + c\), where:

• \(m\) represents the slope of the line, indicating the steepness and direction.
• \(c\) represents the y-intercept, the point where the line crosses the y-axis.

To convert an equation like \(x - 5y = -2\) into slope-intercept form, you need to solve for \(y\). Rearrange the equation: - Start with the original equation: \(x - 5y = -2\)- Subtract \(x\) from both sides: \(-5y = -x - 2\)- Divide everything by \(-5\) to isolate \(y\): \(y = \frac{1}{5}x + \frac{2}{5}\).This process reveals the slope, \(\frac{1}{5}\), and the y-intercept, \(\frac{2}{5}\). This form enables you to easily see and utilize the characteristics of the line.

Parallel Lines

Parallel lines have always been fascinating in geometry. Interestingly, in algebra, identifying parallel lines involves comparing their slopes. Two lines are parallel if they have the same slope. Here's why:

• When lines have the same slope, they rise over run at the same rate, maintaining a consistent distance apart.
• Parallel lines will never intersect because their directions are perfectly aligned.

In our example, lines \(L_{1}\) and \(L_{2}\) were both converted to the form \(y = mx + c\). The slopes were found to be \(\frac{1}{5}\) for both lines. Since they share the same slope, we conclude that \(L_{1}\) and \(L_{2}\) are parallel. This lack of intersection is key in many practical applications like creating parallel roads or railway tracks.

Perpendicular Lines

Perpendicular lines form right angles with each other, a concept that is immensely useful in various fields such as architecture and design. In the context of algebra, two lines are considered perpendicular if their slopes are negative reciprocals. This means:

• If one line has a slope of \(m\), the other will have a slope of \(-\frac{1}{m}\).
• The product of their slopes is \(-1\). For example, if one slope is \(\frac{1}{5}\), the reciprocal would be \(-5\).

In the exercise, even though we calculated the slopes, they were equal and not negative reciprocals. Thus, \(L_{1}\) and \(L_{2}\) are not perpendicular. Recognizing perpendicular lines is vital for ensuring structures are built with precise right angles, a basic necessity in construction and drafting.