Problem 8 - Algebra and Trigonometry

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 through points P and Q is \( \frac{1}{2} \).

Step 1: Understand the Slope Formula

To find the slope of the line through two points, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Assign Point Coordinates to Variables

Identify the coordinates of points P and Q. Here, P is \((1, 2)\) which means \(x_1 = 1\) and \(y_1 = 2\). For point Q \((3, 3)\), \(x_2 = 3\) and \(y_2 = 3\).

Step 3: Substitute Values into Slope Formula

Now substitute these values into the slope formula:\[ m = \frac{3 - 2}{3 - 1} \]

Step 4: Simplify the Expression

Calculate the differences: \(y_2 - y_1 = 3 - 2 = 1\) and \(x_2 - x_1 = 3 - 1 = 2\). The slope is then:\[ m = \frac{1}{2} \]

Key Concepts

Finding the Slope

The slope of a line is an important concept in algebra and geometry. It's a measure of the steepness of the line. Understanding how to find it is essential for graphing and interpreting linear relationships.
To find the slope, you'll use the slope formula:

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

This formula helps you calculate the change in the vertical direction (rise) relative to the change in the horizontal direction (run) between two points on the line.
Once you know the positions of two points, you can calculate the slope by substituting their coordinates into the formula, giving you a clear numeric representation of the line's incline.

Coordinates

Coordinates are pairs of numbers that define the position of a point on a plane. In two-dimensional geometry, each point is expressed as

• o(x, y)

where 'x' represents the horizontal distance and 'y' the vertical distance from a fixed reference point called the origin.
Understanding coordinates is crucial because they provide the necessary information to graph points and lines on a Cartesian plane.
For our task, knowing the coordinates of points P and Q—

• P(1,2)
• Q(3,3)

is vital to find the slope accurately. By identifying these coordinates, you lay the groundwork for applying the slope formula effectively.

Mathematics Formula

Mathematical formulas are powerful tools used to simplify complex calculations into manageable steps. The slope formula is a classic example. It condenses the concept of steepness into an easy-to-use equation.
Here's how we apply the slope formula in practice:

• Identify the coordinates from the points, P(1,2) and Q(3,3).
• Plug these coordinates into the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• Calculate: \[ m = \frac{3 - 2}{3 - 1} = \frac{1}{2} \]

By following these steps, the daunting task of determining the line's slope becomes straightforward.
Mathematical formulas not only streamline difficult concepts but also make calculations efficient and error-free.