Problem 48 - Algebra A Combined Function

Question

Find the slope of the line that is (a) parallel and (b) perpendicular to the line through each pair of points. See Example 7. $$ (6,-1) \text { and }(-4,-10) $$

Step-by-Step Solution

Answer

Parallel slope: \(\frac{9}{10}\). Perpendicular slope: \(-\frac{10}{9}\).

Step 1: Determine the formula for slope

The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We will use this formula to find the slope of the line through the given points.

Step 2: Substitute the points into the slope formula

Substitute the points \((6, -1)\) and \((-4, -10)\) into the slope formula. This means substituting \(x_1 = 6\), \(y_1 = -1\), \(x_2 = -4\), and \(y_2 = -10\).

Step 3: Calculate the slope of the line

Using the formula, calculate \( m = \frac{-10 - (-1)}{-4 - 6} = \frac{-10 + 1}{-10} = \frac{-9}{-10} = \frac{9}{10} \). Therefore, the slope of the line through the given points is \( \frac{9}{10} \).

Step 4: Determine the slope of the parallel line

Lines that are parallel have the same slope. Therefore, the slope of a parallel line to the given line is \( \frac{9}{10} \).

Step 5: Determine the slope of the perpendicular line

Perpendicular lines have slopes that are negative reciprocals of each other. To find the slope of the perpendicular line, take the negative reciprocal of \( \frac{9}{10} \), which is \(- \frac{10}{9} \).

Key Concepts

Parallel Lines

Parallel lines are lines in the plane that never meet, no matter how far they are extended. This means they are always the same distance apart. In terms of slopes, if two lines are parallel, they have exactly the same slope.
For instance, if you calculate the slope of a line and get a result of \( \frac{9}{10} \), any line that is parallel to this one will also have a slope of \( \frac{9}{10} \). This is because parallel lines maintain the same inclination to ensure they never intersect.

• Slopes of parallel lines are equal.
• If one line's slope is \( m \), a parallel line's slope will also be \( m \).
• Being parallel is all about maintaining a uniform distance, which mathematically translates into identical slopes.

So, to find a slope of a line parallel to another, simply look for the same numerical value as the other line's slope.

Perpendicular Lines

Perpendicular lines intersect at a 90-degree angle, forming an "L" shape. This unique intersection property reflects on their slopes as well. If you know the slope of one line, the slope of a perpendicular line will be its negative reciprocal.
Imagine you have a line with a slope of \( \frac{9}{10} \). The slope of a line perpendicular to this one is not just different; it is the negative reciprocal, which turns it into \(- \frac{10}{9} \).

• Perpendicular means intersecting at a right angle.
• The slopes of two perpendicular lines are negative reciprocals.
• If one slope is \( m \), the slope of the perpendicular line is \(-\frac{1}{m}\).

Understanding this relationship helps in detecting perpendicular lines and solving geometry problems that involve right angles.

Negative Reciprocal

A negative reciprocal is a concept tied closely to slopes of perpendicular lines. If you have a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). To find the negative reciprocal, you simply negate the reciprocal, resulting in \( -\frac{b}{a} \).
For slope calculations, especially involving perpendicular lines, this is crucial. So, if you have a slope of \( \frac{9}{10} \), the negative reciprocal—and thus the slope of any perpendicular line—will be \(-\frac{10}{9} \).

• The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
• Add a negative sign to get the negative reciprocal: from \( \frac{b}{a} \) to \(-\frac{b}{a}\).
• Used most importantly for identifying the slopes of perpendicular lines.

Mastery of this concept aids in understanding and solving more complex math problems involving geometric constructs.

Slope Formula

The slope formula is a cornerstone in geometry and algebra, providing the mechanism to calculate the slope of a line using two distinct points on the line. The formula reads \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two points on the line.
This formula calculates how steep the line is by determining the change in vertical direction (\( \Delta y \)) divided by the change in horizontal direction (\( \Delta x \)).

• The slope, \( m \), shows how much \( y \) changes for a unit change in \( x \).
• Positive slope signals a line rising rightward, whereas a negative slope indicates it falls rightward.
• This formula is pivotal for finding parallel and perpendicular relationships in lines.

Using the slope formula, students can analyze and work with lines in various problem scenarios, making it a fundamental tool in analytics and design across physics, engineering, and economics.