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Chapter 2: Problem 78

Determine if the lines \(L_{1}\) and \(L_{2}\) passing through the indicated pairs of points are parallel, perpendicular, or neither. \(L_{1}:(-1,6),(1,4) ; L_{2}:(3,-3),(6,-9)\)

Short Answer

Expert verified
The lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Step by step solution

01

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

To calculate the slope of \(L_{1}\), use the formula for slope which is \( m = (y_{2} - y_{1}) / (x_{2} - x_{1}) \). Here, the coordinates of the two points are (-1,6) and (1,4), so \( m_{1} = (4 - 6) / (1 - (-1)) = -1 \).
02

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

Similarly, calculate the slope of \(L_{2}\). The coordinates of the two points are (3,-3) and (6,-9). Using the slope formula yields \( m_{2} = (-9 - (-3)) / (6 - 3) = -2 \).
03

Compare the slopes

To check if two lines are parallel, their slopes should be equal. Here, \( m_{1} \) is -1 and \( m_{2} \) is -2. So \( L_{1} \) and \( L_{2} \) are not parallel. To check if they are perpendicular, the slopes should be negative reciprocals. However, the negative reciprocal of -1 isn't -2 but 1, so \( L_{1} \) and \( L_{2} \) are not perpendicular. Therefore, \( L_{1} \) and \( L_{2} \) are neither parallel nor perpendicular.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculating Slope
Understanding how to calculate the slope of a line is a cornerstone in algebra, as it expresses how steep a line is. The slope is often represented by the letter 'm' and is calculated by the change in the 'y' values divided by the change in the 'x' values between two points on the line.

To put it into a formula, if we have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope m is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Calculating slope enables us to understand the direction of a line. A positive slope means the line rises from left to right, a negative slope means it falls, and if the slope is zero, the line is horizontal. An undefined or infinite slope corresponds to a vertical line where \( x_1 \) and \( x_2 \) would be equal, leading to division by zero in the slope formula.
Parallel and Perpendicular Lines
In algebra, it's important to recognize when lines are parallel or perpendicular to each other due to their distinct properties. Parallel lines are two or more lines that never intersect. They always have the same slope, and in a graph, they look like they are always the same distance apart.

On the other hand, perpendicular lines intersect at a 90-degree angle, and their slopes are negatively reciprocal of each other. This means if one line has a slope of \( m \), the other line's slope will be \( -\frac{1}{m} \) if the lines are perpendicular. To verify if two lines are perpendicular, you multiply their slopes and check if the product is -1.
Slope Formula
The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is not just a theoretical concept; it's a practical tool for analyzing the characteristics of lines. This formula can give insights into how fast a line is rising or falling and provide critical information to solve real-world problems involving rates of change.

Quick tips for using the slope formula effectively:
  • Always subtract the coordinates in the same order (\((y_2 - y_1)\) and \((x_2 - x_1)\)) to maintain consistency.
  • Calculate the difference in 'y' values (rise) first and 'x' values (run) second to get the proper slope.
  • Remember that horizontal lines have a slope of 0, and vertical lines have an undefined slope.
  • Check your work by plotting the points and visually assessing the rise over run - it should match your calculated slope.

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