Problem 6 - 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 is -3.

Step 1: Identify the coordinates

Identify the coordinates of points \( P \) and \( Q \). Point \( P \) has coordinates \((0,0)\) and point \( Q \) has coordinates \((2,-6)\).

Step 2: Formulate the slope formula

Recall the formula for the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Plug the coordinates into the formula

Substitute \((x_1, y_1) = (0,0)\) and \((x_2, y_2) = (2,-6)\) into the slope formula: \[ m = \frac{-6 - 0}{2 - 0} \]

Step 4: Calculate the differences

Perform the subtraction in the numerator and the denominator: \[ m = \frac{-6}{2} \]

Step 5: Simplify the fraction

Divide the numerator by the denominator to find the slope: \[ m = -3 \]

Step 6: State the result

The slope of the line through points \( P(0,0) \) and \( Q(2,-6) \) is \( -3 \).

Key Concepts

Coordinate Geometry

Understanding coordinate geometry helps us to describe the position of points in a plane using coordinates. Every point is expressed as an ordered pair \(x, y\).
For instance, in this exercise, we have two distinct points: \(P(0,0)\) and \(Q(2,-6)\). These coordinates allow us to map the exact location of each point on a Cartesian plane.
This system provides a way to analyze geometrical shapes and systems algebraically, thereby connecting the two fields of geometry and algebra more closely. The Cartesian plane is divided into four quadrants, allowing you to locate any point with simple measurements of horizontal (x-axis) and vertical (y-axis) distances from the origin.

Slope Formula

The slope of a line is a measure of its steepness and direction in coordinate geometry. You can find the slope between two points using the slope formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
In our case, the coordinates for \(P\) are \((0,0)\) and for \(Q\) are \((2,-6)\). When we plug these into the formula, we calculate:
1. Difference in y-values: \(y_2 - y_1 = -6 - 0 = -6\)
2. Difference in x-values: \(x_2 - x_1 = 2 - 0 = 2\)
The slope becomes \(m = \frac{-6}{2} = -3\).
A negative slope like \(-3\) indicates the line falls or declines as it moves from left to right.

Linear Algebra

Linear algebra is a fundamental part of mathematics, dealing with vectors, spaces, and linear transformations. One of the core concepts in linear algebra is the equation of a line, often written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

• In the given exercise, once we find the slope \(-3\), it helps characterize the entire relationship in a simple linear form.
• Understanding such linear equations allows you to graph the line, predict its behavior, and establish relationships between different data points.
• This foundational knowledge is extensively used in fields such as physics, computer science, and statistics where modeling linear relationships is crucial.

The slope not only tells us the gradient of the line but also affects how we interpret and model data efficiently in a visual and algebraic manner.