Problem 46 - Elementary Algebra Within Reach

Question

In Exercises 46-49, determine whether the lines \(L_{1}\) and \(L_{2}\) that pass through the pairs of points are parallel, perpendicular, or neither. $$ \begin{aligned} &L_{1}:(8,7),(4,4) \\ &L_{2}:(2,1),(-1,5) \end{aligned} $$

Step-by-Step Solution

Answer

Lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

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

The slope of \(L_{1}\) which passes through points (8,7) and (4,4) is obtained by substituting these coordinates into the slope formula: \(m_{1} = (7 - 4) / (8 - 4) = 3 / 4 = 0.75\).

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

Now, calculate the slope of \(L_{2}\) which passes through points (2,1) and (-1,5). We substitute these coordinates into the slope formula: \(m_{2} = (5 - 1) / (-1 - 2) = -4 / -3 = 1.33\).

Step 3: Compare the slopes

In this step, the calculated slopes of \(L_{1}\) and \(L_{2}\) are compared. First, check if the slopes are equal (in which case the lines would be parallel). The slopes \(m_{1} = 0.75\) and \(m_{2} = 1.33\) are not equal, so the lines are not parallel. Also, the product of the slopes \(m_{1}*m_{2} = 0.75*1.33 \neq -1 \), which would have indicated that lines are perpendicular. Therefore, the lines are neither parallel nor perpendicular. Hence, the lines are neither parallel nor perpendicular.

Key Concepts

Slope Calculation

When trying to understand the orientation of a line or the relationship between two lines on a plane, one of the most crucial concepts is the slope calculation. The slope of a line represents its steepness and direction. It is the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. To find this value, we simply select two points on the line, \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) and use the formula:
\[ m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \]
This calculation gives us a single number that helps characterize the line. If the slope is positive, the line goes upwards as it moves from left to right. If it's negative, the line goes downwards. If the slope equals zero, that denotes a horizontal line, and if the slope is undefined (due to a zero denominator), then the line is vertical. The slope is intimately connected with the angle the line makes with the horizontal axis, hence understanding it is key to analyzing the behavior of lines in a coordinate system.

Slope Formula

The slope formula, \( m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \), is a fundamental tool in algebra and geometry that allows us to compute the slope of a line when given two points.

Importance of the Slope Formula

Using the slope formula, we can determine whether two lines are parallel, perpendicular, or neither just by looking at numerical values. It is essential for finding the equation of a line, predicting the points it will pass through, and understanding its behavior. In practical terms, it is used in fields such as engineering, physics, and graphics design to comprehend the relationships between various lines in a system or design.

Application in Word Problems and Graphs

When applying the slope formula, one must carefully substitute the given coordinates of the two points correctly to avoid any arithmetic errors. In the context of word problems or interpreting graphs, calculating the slope provides insights into rates of change, allowing us to make predictions and understand patterns.

Comparing Slopes

The final step in understanding the relationship between two lines often involves comparing their slopes. Lines with

• the same slope are parallel;
• a product of their slopes equaling -1 are perpendicular;
• and slopes with neither of these characteristics are simply intersecting lines.

Let's consider how to compare the slopes of two lines, \(L_{1}\) and \(L_{2}\), which have different slopes calculated as \(m_{1}\) and \(m_{2}\), respectively.

For the lines to be parallel, their slopes must be equal, \(m_{1} = m_{2}\). If they are to be perpendicular, their slopes must be negative reciprocals of each other, meaning their product should be -1, \(m_{1} * m_{2} = -1\). When comparing slopes that do not meet either condition, we conclude that the lines intersect at some point and do so at an angle that is neither a right angle nor are the lines never meeting, indicating that they are neither perpendicular nor parallel.