Problem 8 - Precalculus Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) $$P(1,2), Q(3,3)$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Slope Formula

The slope of a line through two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is determined by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula calculates how much the \(y\)-value changes for a one-unit change in the \(x\)-value.

Step 2: Identify the Coordinates

Identify the coordinates from the problem. The points given are \(P(1, 2)\) and \(Q(3, 3)\). So, \(x_1 = 1\), \(y_1 = 2\), \(x_2 = 3\), and \(y_2 = 3\).

Step 3: Substitute Coordinates into the Slope Formula

Substitute the coordinates into the slope formula: \[ m = \frac{3 - 2}{3 - 1} \].

Step 4: Calculate the Slope

Perform the subtraction for both the numerator and the denominator in the fraction: \(3 - 2 = 1\) and \(3 - 1 = 2\). Thus, the slope \(m\) is \( \frac{1}{2} \).

Key Concepts

Slope Formula

To determine the slope of a line, we use the slope formula. It's a mathematical formula that helps us understand how steep a line is. This formula is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where:

• \( m \) stands for slope.
• \( x_1, y_1 \) are the coordinates of the first point.
• \( x_2, y_2 \) are the coordinates of the second point.

The formula finds the vertical change (difference in \( y \)-coordinates) relative to the horizontal change (difference in \( x \)-coordinates). A higher slope value means a steeper line, while a lower value indicates a gentler slope.
It's crucial for understanding how points are connected in geometry.

Coordinate Geometry

Coordinate geometry is the study of geometry using a coordinate plane. On this plane, every point is defined by a pair of numbers, known as coordinates. These coordinates help us easily describe the location of points and lines.
The points you see in coordinate geometry are usually denoted by pairs like \((x, y)\). Here, \( x \) refers to the position along the horizontal axis, and \( y \) refers to the position along the vertical axis.This makes coordinate geometry a powerful tool for visualizing and solving problems involving lines. By using coordinates, we can calculate the distance between points, determine whether lines are parallel or perpendicular, and, most importantly, calculate the slope.
It's a bridge between algebra and geometry, allowing for algebraic manipulation of geometric shapes.

Calculating Slope

Calculating the slope between two points is a straightforward process if you follow a few simple steps. Let's apply this to the points \( P(1, 2) \) and \( Q(3, 3) \).

• **Identify the coordinates:** Begin by noting your points, which are \( P(1, 2) \) and \( Q(3, 3) \). This gives us \( x_1 = 1 \), \( y_1 = 2 \), \( x_2 = 3 \), and \( y_2 = 3 \).
• **Substitute into the slope formula:** Place these numbers into the slope formula: \[ m = \frac{3 - 2}{3 - 1} \]
• **Perform the calculations:** Calculate the difference in \( y \) values, \( 3 - 2 = 1 \), and \( x \) values, \( 3 - 1 = 2 \).
• **Divide to find the slope:** Finally, divide the differences. The slope \( m \) is \( \frac{1}{2} \).

This method highlights how much the line rises or falls as you move from one point to another along the x-axis.