Problem 6 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The slope is \(-\frac{4}{3}\).

Step 1: Understand the Slope Formula

The slope of a line passing through two points, \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points.

Step 2: Identify the Coordinates

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

Step 3: Substitute Coordinates into Slope Formula

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

Step 4: Simplify the Formula

Calculate the numerator and denominator separately: Numerator: \( 3 - (-5) = 3 + 5 = 8 \) Denominator: \( -4 - 2 = -6 \). Thus, \( m = \frac{8}{-6} \).

Step 5: Simplify the Slope

Simplify the fraction \( \frac{8}{-6} \) by dividing both the numerator and the denominator by 2: \( \frac{8}{-6} = \frac{4}{-3} \). Thus, the slope \( m = -\frac{4}{3} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, bridges algebra and geometry using a system of coordinates. It allows us to describe geometric shapes such as lines, curves, and polygons through equations and inequalities. In coordinate geometry, each point in the plane is identified by its coordinates, typically written as \(x, y\).

The basic idea is to use the Cartesian coordinate system, which consists of two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical). Any location in this plane is determined by how far along each axis the point is located. This forms the foundation for plotting points and interpreting slopes, lines, and shapes.

Two main applications of coordinate geometry include:

• Finding the distance between two points using the distance formula.
• Determining the slope of a line, which we will explore further!

Slope of a Line

The slope of a line in coordinate geometry quantifies its steepness and direction. It's an essential concept for understanding linear equations and determining how changes in one variable affect another.

The slope is represented by the symbol \m\ and can be calculated using the slope formula, which is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(x_1, y_1\) and \(x_2, y_2\) represent two distinct points on the line. This formula tells us how much the y-coordinate (or vertical position) changes for a unit change in the x-coordinate (or horizontal position).

Some important facts about the slope:

• A positive slope means the line rises from left to right.
• A negative slope means the line falls from left to right.
• A zero slope indicates a horizontal line.
• An undefined slope (division by zero) indicates a vertical line.

When calculating the slope between two points, like points \(P(2,-5)\) and \(Q(-4,3)\), substituting these coordinates into the slope formula helps find the slope efficiently.

Simplifying Fractions

Simplifying fractions is an important skill in mathematics that makes expressions easier to understand and work with. When you have a fraction, simplifying it means making it as simple as possible by finding the greatest common factor (GCF) of the numerator and the denominator, then dividing both by this number.

In our exercise, after calculating the slope as \(\frac{8}{-6}\), we simplify by dividing both 8 and -6 by their GCF, which is 2:\[ \frac{8}{-6} = \frac{8 \div 2}{-6 \div 2} = \frac{4}{-3} \]This gives us the simplified slope of \-\frac{4}{3}\. It's crucial to simplify fractions, as it provides a clearer, more precise answer, which is especially useful in coordinate geometry and algebra.

A few tips when simplifying fractions:

• Always look for the largest number that evenly divides both parts of the fraction.
• If the numerator and denominator share no common factors other than 1, the fraction is already simplified.
• Simplifying helps with calculations and comparisons in math problems.