Problem 45 - College Algebra

Question

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. \((-1,3)\) and \((5,1)\) \((-2,3)\) and \((0,9)\)

Step-by-Step Solution

Answer

The lines are perpendicular with slopes of \\(-\frac{1}{3}\\) and \\(3\\).

Step 1: Identify Coordinates

The given points for the first line are \((-1,3)\) and \(5,1)\). The points for the second line are \((-2,3)\) and \(0,9)\).

Step 2: Calculate Slope of the First Line

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points \((-1,3)\) and \(5,1)\). Substitute the coordinates: \\[m = \frac{1 - 3}{5 - (-1)} = \frac{-2}{6} = -\frac{1}{3}\].

Step 3: Calculate Slope of the Second Line

Use the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\) for points \((-2,3)\) and \(0,9)\). Substitute the coordinates: \\[m = \frac{9 - 3}{0 - (-2)} = \frac{6}{2} = 3\].

Step 4: Compare Slopes and Determine Relationship

For the first line, the slope is \(-\frac{1}{3}\) and for the second line, it is \(3\). Since the product of the slopes \(-\frac{1}{3} \times 3 = -1\), these lines are perpendicular.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of geometry where points are defined and labeled using an ordered pair of numbers, known as coordinates. This is a very important branch of mathematics because it combines the features of algebra with geometry to describe geometric shapes through equations.

In coordinate geometry, every point is identified by an

• x-coordinate (the horizontal position)
• y-coordinate (the vertical position).

These coordinates are usually written as \( (x, y) \). Any equation involving these coordinates can describe a shape or the relation between points in a plane. It allows us to work with geometric problems using algebraic techniques, offering a powerful tool to solve complex mathematical problems efficiently.

Using the coordinate system, you can calculate various properties:

• Distance between two points
• Midpoint of a line segment
• Slopes of the lines

Mastering coordinate geometry opens the door to understanding more complex geometric concepts and solving numerous real-world problems.

Parallel and Perpendicular Lines

In coordinate geometry, understanding parallel and perpendicular lines is essential to recognizing relationships between different lines.

**Parallel Lines** are lines in a plane that never meet. They have the same slope or gradient. This means if you calculate the slope of two lines and find them to be equal, the lines are parallel. For example, lines with slopes \(m_1 = 2\) and \(m_2 = 2\) are parallel because their slopes are identical.

**Perpendicular Lines**, on the other hand, intersect at a right angle (90 degrees). When two lines are perpendicular, the product of their slopes is \-1\. This means if you have two lines with slopes \(m_1\) and \(m_2\), they are perpendicular if \(m_1 \times m_2 = -1\).

This knowledge quickly assists in determining relationships using slopes.

• Calculate each line's slope.
• Compare for equality (for parallel lines).
• Check if the product equals \-1\ (for perpendicular lines).

It simplifies analyzing line relationships in both academic and practical geometry problems.

Equation of a Line

The equation of a line in coordinate geometry can describe every point lying on the line, and it's one of the most fundamental concepts in mathematics. There are several forms of equations for a line, but the slope-intercept form is quite popular for its simplicity.

In slope-intercept form, the equation is written as \(y = mx + c\), where:

• \(m\) is the slope of the line.
• \(c\) is the y-intercept, which is the point where the line crosses the y-axis.

By knowing the slope and one point on the line, you can find the entire equation of the line. The slope describes how steep the line is, while the intercept serves to position the line vertically on a graph.

Another common form is the point-slope form: \(y - y_1 = m(x - x_1)\), especially useful for deriving the equation if you know the slope and one point on the line. You can easily transform between these forms to suit different algebraic scenarios.

Understanding these equations helps greatly in graphing lines, analyzing linear relationships, and solving problems involving slopes and intercepts.