Problem 10 - Precalculus Mathematics for Calculus

Question

Find the slope of the line through \(P\) and \(Q .\) $$P(2,-5), Q(-4,3)$$

Step-by-Step Solution

Answer

The slope of the line through points \( P \) and \( Q \) is \(-\frac{4}{3}\).

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 given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \( m \) is the slope of the line.

Step 2: Identify the Coordinates

Identify the coordinates of the points \( P \) and \( Q \). For \( P \), \( x_1 = 2 \) and \( y_1 = -5 \). For \( Q \), \( x_2 = -4 \) and \( y_2 = 3 \).

Step 3: Substitute into the Slope Formula

Substitute the coordinates into the slope formula:\[ m = \frac{3 - (-5)}{-4 - 2} \]

Step 4: Simplify the Numerator and Denominator

Calculate the changes in \( y \) and \( x \):- Numerator: \( 3 - (-5) = 3 + 5 = 8 \)- Denominator: \( -4 - 2 = -6 \)So, the fraction becomes \( \frac{8}{-6} \).

Step 5: Simplify the Slope

Simplify the fraction \( \frac{8}{-6} \) by dividing both numerator and denominator by their greatest common divisor (2):\[ m = \frac{8 \div 2}{-6 \div 2} = \frac{4}{-3} \]Therefore, the slope of the line is \( -\frac{4}{3} \).

Key Concepts

Slope Formula

The slope formula is a crucial concept in coordinate geometry, as it helps us determine the steepness or inclination of a line connecting two points on a plane. This formula is expressed as:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

where:

• \( m \) is the slope.
• \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line.

It's important to understand that the slope represents the "rise over run," or how much the line rises vertically for every unit it runs horizontally. If the slope is positive, the line ascends as you move from left to right. If it's negative, the line descends. This formula allows us to quantify alignment and gradient, making it easier to graph and analyze lines.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful tool for studying geometric figures algebraically. By assigning coordinates to points, we can use equations to represent geometric shapes like lines, circles, and parabolas.
In this context, understanding the slope of a line becomes essential. The slope is a number that tells us the direction and steepness of a line in a coordinate plane.
With the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), you can determine how one variable changes with respect to another. This aligns with the broader scope of coordinate geometry, where relationships between points and lines can be examined through equations and graphs. The beauty of coordinate geometry lies in its ability to blend algebra with geometry, providing a framework for solving complex problems with precision.

Points on a Plane

Points on a plane are the foundational building blocks of coordinate geometry. Each point is represented by a pair of numbers, known as coordinates. These coordinates are usually written as \((x, y)\) where \(x\) denotes the horizontal position, and \(y\) denotes the vertical position on a Cartesian plane.
In the problem, we consider the points \(P(2, -5)\) and \(Q(-4, 3)\).

• For \(P\), the coordinates are \(x_1 = 2\) and \(y_1 = -5\).
• For \(Q\), the coordinates are \(x_2 = -4\) and \(y_2 = 3\).

When examining points like these on a plane, we can determine various geometric properties such as the distance between points, and more relevantly here, the slope of the line connecting these points. Recognizing the way coordinates map out on a plane gives us the advantage of visualizing and interpreting spatial relationships, thus deepening our understanding of geometry in a practical and applicable manner.