Problem 2 - Algebra and Trigonometry

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(0,0), Q(2,-6)\)

Step-by-Step Solution

Answer

The slope of the line through points \( P \) and \( Q \) is \(-3\).

Step 1: Understand the Formula for Slope

The slope of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula helps us determine how much \( y \) changes per unit change in \( x \).

Step 2: Identify the Coordinates of the Points

The coordinates for point \( P \) are \( (0,0) \), and the coordinates for point \( Q \) are \( (2,-6) \). Here, \( x_1 = 0 \), \( y_1 = 0 \), \( x_2 = 2 \), and \( y_2 = -6 \).

Step 3: Substitute the Values into the Slope Formula

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), substitute the coordinates: \( m = \frac{-6 - 0}{2 - 0} \).

Step 4: Simplify the Expression to Find the Slope

Simplify the expression \( \frac{-6}{2} \) to find the slope. When simplified, \( \frac{-6}{2} = -3 \).

Key Concepts

Coordinate Geometry

In the vast world of mathematics, coordinate geometry serves as a fundamental bridge between algebra and geometry. It uses a coordinate system to define every point in a plane using ordered pairs.

This pair, known as coordinates, consists of an *x*-coordinate and a *y*-coordinate, typically represented as \((x, y)\). This system empowers us to pinpoint locations and describe geometric shapes precisely.

Coordinate geometry is particularly useful because it allows us to perform algebraic calculations to solve geometric problems. By examining relationships between points in a plane, such as lines or curves, we can apply algebraic methods for deeper analysis. This approach makes it much easier to visualize abstract mathematical concepts.

For instance, once points like \(P(0, 0)\) and \(Q(2, -6)\) are defined within this coordinate system, one can perform calculations to understand the line connecting them, among other relationships - such as the distance between the points or the slope of the line.

Slope Formula

The slope of a line is a critical concept within coordinate geometry. It represents how steep or flat a line is, which is typically expressed as the ratio of the vertical changes to the horizontal changes between two points on a line.

The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the slope \(m\) using two points with coordinates \((x_1, y_1)\) and \((x_2, y_2)\). Here,

• \(y_2 - y_1\) is the change in the vertical direction
• \(x_2 - x_1\) is the change in the horizontal direction

Understanding this can help us discern not only the incline of a line but also if it's increasing or decreasing. A positive slope indicates an upward incline, while a negative slope means the line descends. A zero slope suggests a flat, horizontal line, and an undefined slope reflects a vertical line.

In the case of points \(P(0, 0)\) and \(Q(2, -6)\), using the formula, the slope is:
\[ m = \frac{-6 - 0}{2 - 0} = \frac{-6}{2} = -3 \]Thus, the line through these points descends as you move from left to right.

Point Coordinates

Point coordinates are essential for plotting points and solving problems in coordinate geometry. A point on a Cartesian plane is defined by its coordinates, which are always written in an ordered pair format \((x, y)\).

In our given example, the coordinates for point \(P\) are \((0, 0)\) and for point \(Q\) are \((2, -6)\). Here,

• The first number \(x\) in each pair specifies the horizontal position.
• The second number \(y\) specifies the vertical position.

Each coordinate provides essential information:

• \(P(0,0)\) is at the origin, where both axes intersect, representing the starting point.
• \(Q(2,-6)\) represents a point two units to the right and six units below the origin.

By correctly identifying these coordinates, you can effectively apply the slope formula, analyze line segments, or even construct more complex geometrical shapes. This clarity in numerical terms translates the visual aspect of geometry into a form that can be calculated and analyzed through algebra.