Problem 5 - Algebra and Trigonometry

Question

Find the slope of the line through P and Q. $$ P(0,0), Q(4,2) $$

Step-by-Step Solution

Answer

The slope of the line is \( \frac{1}{2} \).

Step 1: Identify the Coordinates

First, identify the coordinates of the points through which the line passes. The coordinates given are \( P(0,0) \) and \( Q(4,2) \). Let point \( P \) have coordinates \( (x_1, y_1) \) and point \( Q \) have coordinates \( (x_2, y_2) \). Thus, \( x_1 = 0 \), \( y_1 = 0 \), \( x_2 = 4 \), and \( y_2 = 2 \).

Step 2: Use the Slope Formula

The formula to calculate the slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Replace \( (x_1, y_1) \) with \( (0, 0) \) and \( (x_2, y_2) \) with \( (4, 2) \).

Step 3: Substitute the Coordinates in the Formula

Substitute the known values into the slope formula: \[ m = \frac{2 - 0}{4 - 0} = \frac{2}{4} \]

Step 4: Simplify the Fraction

Simplify the fraction \( \frac{2}{4} \) to its lowest terms: \[ \frac{2}{4} = \frac{1}{2} \] Thus, the slope of the line is \( \frac{1}{2} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that deals with points, lines, and shapes in a flat plane. It uses a coordinate system to describe the position of points on a plane. In coordinate geometry, each point is defined by a pair of numbers known as coordinates. These coordinates are usually written as \((x, y)\). The first number \(x\) is the horizontal position, and the second number \(y\) is the vertical position.
By understanding the coordinates of points like \(P(0,0)\) and \(Q(4,2)\), you can determine the relationships between points, lines, and shapes. In this exercise, coordinate geometry allows us to understand how the line connects points \(P\) and \(Q\) and to calculate its slope using these coordinates.
Recognizing this concept equips you with the ability to work with lines and shapes on an X and Y plane, where each point has its specific location in the plane.

Slope Formula

The slope formula is a key concept in coordinate geometry used to determine the steepness or incline of a line connecting two points. The slope \(m\) of a line is calculated using the coordinates of two distinct points on the line. This is given by the formula:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula calculates how much the line rises or falls for each unit of horizontal change. For the given points \((0,0)\) and \((4,2)\), we calculate the change in the \(y\)-coordinates (\(y_2 - y_1\)) and the change in the \(x\)-coordinates (\(x_2 - x_1\)).
By subtracting the \(y\)-coordinates and the \(x\)-coordinates separately, we determine the rise and run of the line. Plugging these values into the formula provides the slope of the line, allowing us to describe its steepness effectively.
The slope can tell us whether the line is increasing, decreasing, or constant. A positive slope like \(\frac{1}{2}\) indicates the line is rising, while a negative slope would indicate it is falling.

Simplifying Fractions

Simplifying fractions is an essential mathematical skill that makes working with numbers easier by reducing them to their simplest form. When you have a fraction like \(\frac{2}{4}\), simplifying means finding the greatest common divisor (GCD) of the numerator (top number) and the denominator (bottom number) and dividing both by this number.
In this exercise, the original fraction for the slope \(\frac{2}{4}\) simplifies to \(\frac{1}{2}\). This is done by recognizing that both 2 and 4 can be divided by 2. Once simplified, the fraction \(\frac{1}{2}\) provides a clearer understanding of the slope without changing its value, hence making it more usable in calculations.
Simplifying fractions is crucial in not just this context but in various mathematical problems as it often makes calculations easier to perform and interpret. It's a simple yet powerful tool for making numbers cleaner and more understandable.