Problem 9 - Precalculus Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) $$P(2,4), Q(4,3)$$

Step-by-Step Solution

Answer

The slope of the line through \(P\) and \(Q\) is \(-\frac{1}{2}\).

Step 1: Understand the slope formula

The slope of a 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 calculates the rate of change of \(y\) with respect to \(x\).

Step 2: Identify the coordinates

The coordinates of the points are \(P(2, 4)\) and \(Q(4, 3)\). Here, \((x_1, y_1) = (2, 4)\) and \((x_2, y_2) = (4, 3)\).

Step 3: Substitute into the formula

Substitute the coordinates of the points into the slope formula: \( m = \frac{3 - 4}{4 - 2} \).

Step 4: Calculate the difference in \(y\)-coordinates

Find the difference in the \(y\)-coordinates: \(3 - 4 = -1\).

Step 5: Calculate the difference in \(x\)-coordinates

Find the difference in the \(x\)-coordinates: \(4 - 2 = 2\).

Step 6: Calculate the slope

Divide the difference in \(y\)-coordinates by the difference in \(x\)-coordinates: \(m = \frac{-1}{2}\). The slope of the line through points \(P\) and \(Q\) is \(-\frac{1}{2}\).

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. We use the slope formula to calculate this value, which is fundamental in understanding linear relationships in math. The formula for the slope \( m \) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula helps us determine how much the \( y \)-coordinate (vertical change) changes for every unit change in the \( x \)-coordinate (horizontal change).
This is known as the rate of change of the line, and it tells us whether the line is moving upwards or downwards as we read it from left to right. A positive slope indicates an increasing line, while a negative slope indicates a decreasing line.

Rate of Change

In the context of a line, the rate of change refers to how fast one variable changes in relation to another. The slope of the line essentially measures this rate of change. It shows the relationship between the vertical changes compared to horizontal changes, which can also be interpreted as the rise over the run.

When you calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), you are computing how much the \( y \)-position changes as the \( x \)-position changes:

• If the slope is positive, \( y \) is increasing as \( x \) increases, meaning the line is going upwards.
• If the slope is negative, \( y \) is decreasing as \( x \) increases, meaning the line is going downwards.

This concept is pivotal in many real-world applications, like determining the speed of an object or understanding trends over time.

Coordinates of Points

Coordinates are pairs of numbers that define a specific location on a two-dimensional plane. In mathematics, points like \( P(2,4) \) and \( Q(4,3) \) are examples of how coordinates are used to specify positions:

• The first number in the pair is known as the \( x \)-coordinate, which explains the horizontal position.
• The second number is known as the \( y \)-coordinate, which describes the vertical position.

Understanding how to read these coordinates is essential when working with the slope formula. It allows us to identify the position of points on a graph, making it possible to analyze and calculate the slope between these points.
Knowing how to handle coordinates effectively is a key skill in plotting and understanding graphs.

Difference in Coordinates

When finding the slope of a line between two points, the concept of differences in coordinates is crucial. We use the differences between the coordinates to calculate the slope. Specifically, these are the differences in the \( y \)-coordinates and the \( x \)-coordinates of the points.

For points \((x_1, y_1)\) and \((x_2, y_2)\), the differences are calculated as:

• Difference in \( y \)-coordinates: \( y_2 - y_1 \)
• Difference in \( x \)-coordinates: \( x_2 - x_1 \)

In the solution given, the difference in \( y \)-coordinates is \( 3 - 4 = -1 \) and the difference in \( x \)-coordinates is \( 4 - 2 = 2 \).
By understanding these differences, you can effectively apply the slope formula to determine the line's slope, ensuring accurate calculations.