Problem 47 - 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. $$ (-8,-4) \text { and }(3,5) $$

Step-by-Step Solution

Answer

Parallel slope: \(\frac{9}{11}\); Perpendicular slope: \(-\frac{11}{9}\).

Step 1: Determine the slope of the given line

To find the slope of the line through the points \((-8,-4)\) and \((3,5)\), use the slope formula \( m = \frac{y_2-y_1}{x_2-x_1} \). Substituting the given points, we have \( x_1 = -8 \), \( y_1 = -4 \), \( x_2 = 3 \), and \( y_2 = 5 \). Calculate \( m = \frac{5 - (-4)}{3 - (-8)} = \frac{5 + 4}{3 + 8} = \frac{9}{11} \). So, the slope of the line is \( \frac{9}{11} \).

Step 2: Find the slope of a line parallel to the given line

Lines that are parallel have the same slope. Therefore, the slope of the line parallel to the line through points \((-8,-4)\) and \((3,5)\) is also \( \frac{9}{11} \).

Step 3: Find the slope of a line perpendicular to the given line

The slope of a line perpendicular to another is the negative reciprocal of the original slope. The negative reciprocal of \( \frac{9}{11} \) is \( -\frac{11}{9} \). Therefore, the slope of a line perpendicular to the line through points \((-8,-4)\) and \((3,5)\) is \( -\frac{11}{9} \).

Key Concepts

Parallel Lines and Their Slopes

Parallel lines are fascinating because they maintain a consistent relationship with each other. Think of railroad tracks that keep running side by side without ever crossing. In the world of algebra, parallel lines share an essential trait: they have the same slope. This means no matter how the lines are drawn on the graph, as long as they are not vertical lines they will never intersect.

• The slope determines the steepness and direction of a line.
• For two lines to be parallel, their slopes must be identical.
• This characteristic allows us to quickly determine if lines are parallel, without needing to graph them.

In the case of the line passing through the points e.g., \((-8,-4)\) and \((3,5)\), its slope is calculated as \(\frac{9}{11}\).Any line parallel to this one will also have a slope of \(\frac{9}{11}\). This keeps the two lines equidistant at all points.

Perpendicular Lines and Understanding Negative Reciprocals

Unlike parallel lines, perpendicular lines intersect at a right angle. This creates an exciting relationship between their slopes, involving negative reciprocals.

• A perpendicular slope is the opposite sign and flipped fraction of the original slope.
• This means if a line has a slope of \(m\), a perpendicular line will have a slope of \(-\frac{1}{m}\).
• Understanding this concept helps in solving geometric problems and analyzing angles between lines.

For example, with a line that has a slope of \(\frac{9}{11}\),the perpendicular slope will be \(-\frac{11}{9}\). If you visualize these lines on a graph, you can quite literally see them forming a 'plus' or 'cross' shape, emphasizing their unique intersecting nature.

The Slope Formula and How to Use It

The slope formula is a mathematical tool that allows us to determine the slope of a line given two points. This formula represents the ratio of the vertical change (rise) to the horizontal change (run) between two specific points on a line. Here’s a simple breakdown:

• The formula is \( m = \frac{y_2-y_1}{x_2-x_1} \).
• '\(y_2\)' and '\(y_1\)' represent the y-coordinates of the two points, while '\(x_2\)' and '\(x_1\)' are the x-coordinates.
• You subtract the y-coordinates and x-coordinates to find the change in y and x.

By substituting the values from points like \((-8,-4)\) and \((3,5)\), you plug into the formula:\(m = \frac{5 - (-4)}{3 - (-8)} = \frac{9}{11}\). It's straightforward and provides the slope, assisting in identifying lines' behaviors.Using this powerful formula not only gives clarity on the line’s current angle but also helps in assessing parallel or perpendicular relationships. This makes the slope formula an essential tool in understanding and creating linear graphs.