Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-5,0),(-2,1) ; L_{2}:(0,1),(3,2)\)

Step-by-Step Solution

Answer

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

Step 1: Calculation of the Slope of Line \(L_{1}\)

The formula for the slope is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Plugging in the points of \(L_{1}\) in this formula, the slope \(m_{1}\) of \(L_{1}\) is calculated as \(m_{1}=\frac{1-0}{-2-(-5)} =\frac{1}{3}\)

Step 2: Calculation of the Slope of Line \(L_{2}\)

Similarly, by plugging in the points of \(L_{2}\) in the slope formula, the slope \(m_{2}\) of \(L_{2}\) is calculated as \(m_{2}=\frac{2-1}{3-0} =\frac{1}{3}\)

Step 3: Comparision of Slopes

Both slopes are equal (\(m_{1}=m_{2}=\frac{1}{3}\)). Therefore, the lines \(L_{1}\) and \(L_{2}\) are parallel to each other.

Key Concepts

Slope Calculation

Understanding the concept of slope is fundamental in algebra and geometry. It essentially tells us how steep a line is. To calculate the slope, we often use the formula
\( m=\frac{y_2-y_1}{x_2-x_1} \).
In simpler terms, the slope is the rise over the run, or the change in the vertical direction divided by the change in the horizontal direction between two points on a line.
For example, in the exercise provided, we calculated the slope of line \( L_1 \) using the points (-5,0) and (-2,1). Applying the formula,
\( m_1=\frac{1-0}{-2-(-5)} =\frac{1}{3} \).
The concept can be confusing at first, but remember: the numerator represents how much the line goes up or down as you move from left to right, and the denominator shows how far you move horizontally. A positive slope means the line is ascending; a negative slope means it's descending.

Parallel Lines

When we talk about parallel lines in geometry, we refer to two lines that, no matter how far they extend, will never intersect. The key to determining if two lines are parallel is comparing their slopes. If two lines have the same slope, they are guaranteed to be parallel.
For instance, the slopes of lines \( L_1 \) and \( L_2 \) in our exercise are both equal to \( \frac{1}{3} \). This directly indicates that the two lines are parallel to each other. It's important to remember that parallel lines will always have the same degree of steepness which is why their slopes match.

Perpendicular Lines

Perpendicular lines are quite the opposite of parallel lines. They intersect at a right angle, creating an