Problem 9 - Algebra and Trigonometry

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: Identify the Points

We are given two points, \( P(2,4) \) and \( Q(4,3) \). These are the coordinates we will use to find the slope of the line connecting them.

Step 2: Recall the Slope Formula

The formula for the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We will use this formula to find the slope between points \( P \) and \( Q \).

Step 3: Substitute the Point Values

Substitute \( x_1 = 2 \), \( y_1 = 4 \), \( x_2 = 4 \), and \( y_2 = 3 \) into the slope formula: \( m = \frac{3 - 4}{4 - 2} \).

Step 4: Calculate the Difference in Y-Coordinates

Compute the difference in the y-coordinates: \( y_2 - y_1 = 3 - 4 = -1 \).

Step 5: Calculate the Difference in X-Coordinates

Compute the difference in the x-coordinates: \( x_2 - x_1 = 4 - 2 = 2 \).

Step 6: Compute the Slope

Insert the differences into the slope formula: \( m = \frac{-1}{2} \). Thus, the slope of the line through points \( P \) and \( Q \) is \(-\frac{1}{2}\).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that helps us easily locate points on a plane using an ordered pair of numbers called coordinates. These coordinates identify a point's exact location by assigning a value along the x-axis and y-axis. Usually, the x-coordinate indicates the horizontal position, while the y-coordinate shows the vertical position.
For example:

• In the point \( P(2,4) \), 2 is the x-coordinate representing horizontal movement.
• 4 is the y-coordinate representing vertical movement.

Coordinate geometry can be very useful for visualizing data. It plays a crucial role in various real-world applications, such as engineering, architecture, and physics. By plotting and connecting many such points, you can draw shapes, graphs, and maps.

Mathematical Formula

Formulas are essential tools in mathematics that allow us to find relationships between different quantities. In coordinate geometry, one important formula is the slope formula. It helps us determine the steepness or incline of a line joining two points by calculating the ratio of the change in the vertical direction (rise) to the change in the horizontal direction (run).
The slope formula is given by:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Where

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

This formula can be used in many scenarios beyond simple coordinate geometry, such as finding rates of changes or predicting future values in fields like economics and biology.

Slope Calculation

The calculation of slope involves straightforward steps when using the slope formula. Consider these points: \( P(2,4) \) and \( Q(4,3) \). By following a simple process, we can find the slope of the line passing through them.
Steps to Calculate the Slope:

• Identify and note the coordinates of both points.
• Substitute these coordinates into the formula, \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
• Compute the difference in the y-coordinates: \( 3 - 4 = -1 \).
• Compute the difference in the x-coordinates: \( 4 - 2 = 2 \).
• Insert these results into the formula, yielding \( m = \frac{-1}{2} \).

The final result is a slope of \(-\frac{1}{2}\), which means the line descends as it extends from left to right. The slope is crucial in determining the behavior of the line and predicting how it moves across the coordinate plane.