Question

Slopes of Parallel and Perpendicular Lines Find the slopes of the lines parallel to, and perpendicular to, each line with the given slope. $$m=5$$

Short answer

The slope of lines parallel to the given line is 5, and the slope of lines perpendicular to the given line is -1/5.

Step-by-step solution

Identify the slope of the parallel line

The slope of a line parallel to another line is equal to the slope of the original line. Given that the slope of the original line is m=5, the slope of a parallel line will also be 5.

Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. Thus, the slope of the perpendicular line to the line with slope m=5 is m=-1/5.

Key concepts

Slope of a Line

Understanding the slope of a line is fundamental in algebra and geometry. A slope measures the steepness, incline, or grade of a line. Mathematically, the slope is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two distinct points on the line. If \( (x_1, y_1) \) and \( (x_2, y_2) \) are two points on a line, the slope \( m \) is calculated by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1}\].

If the slope is positive, the line rises from left to right. Conversely, if the slope is negative, the line falls from left to right. Horizontal lines have a slope of zero because there is no vertical change, whereas vertical lines have an undefined slope due to no horizontal change.

Parallel Lines

Parallel lines never intersect and are always the same distance apart. They have an important characteristic that's of particular interest when studying geometrical shapes and algebraic equations: they have the same slope. To determine if two lines are parallel, simply compare their slopes. If the slopes are identical, the lines are parallel.

When working with equations of lines in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, two lines with the same value for \( m \) are guaranteed to be parallel, regardless of the y-intercept value. For example, lines with equations \( y = 5x + 2 \) and \( y = 5x - 3 \) are parallel since their slopes, \( m \) , are both equal to 5.

Perpendicular Lines

Perpendicular lines intersect at a right angle (\(90^\circ\)). The slopes of two perpendicular lines have a very special relationship: they are negative reciprocals of each other. This means that if one line has a slope of \( a \), the other will have a slope of \( -\frac{1}{a} \) if it is perpendicular. To quickly identify the slope of a line that is perpendicular to another, you can use this negative reciprocal relationship.

When calculating the negative reciprocal, remember to invert the slope value and change its sign. For example, consider the slope \( m = 5 \). Its negative reciprocal is \( m = -\frac{1}{5} \) because you flip \( 5 \) to \( \frac{1}{5} \) and then change the sign to negative.

Negative Reciprocal

The concept of negative reciprocal is vital when dealing with perpendicular lines. The negative reciprocal of a number is simply the inverse of the number with an opposite sign. If you have a fraction, you would invert the fraction (i.e., swap the numerator and denominator) before applying the negative sign, while for an integer or a whole number, you would consider it as being over 1.

Illustratively, the negative reciprocal of 5, which we can write as \( \frac{5}{1} \), would be \( -\frac{1}{5} \) and that of \( -\frac{3}{4} \) would be \( \frac{4}{3} \) as you invert and negate it. This principle is a key component in determining the equation of a line perpendicular to a given line.