Problem 2 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(9,10),(6,2)$$

Step-by-Step Solution

Answer

The slope of the line is \(\frac{8}{3}\).

Step 1: Identify the Points

The given points are \((x_1, y_1) = (9, 10)\) and \((x_2, y_2) = (6, 2)\). We will use these coordinates to find the slope of the line.

Step 2: Understand the Slope Formula

The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Substitute the Coordinates into the Slope Formula

Insert the values from the points into the slope formula: \(m = \frac{2 - 10}{6 - 9}\).

Step 4: Calculate the Differences

Calculate the differences in the numerator and denominator: \(m = \frac{-8}{-3}\).

Step 5: Simplify the Slope

Simplify the fraction: \(m = \frac{8}{3}\). This is the slope of the line.

Key Concepts

Coordinate Geometry

In coordinate geometry, every point on a plane can be represented by a pair of numbers known as coordinates. These numbers describe the location of the point relative to the two perpendicular axes on a graph: the x-axis (horizontal) and the y-axis (vertical).
Understanding this concept is essential when dealing with problems that involve the position of points, like finding the slope of a line connecting two points. The coordinates in the first point (9, 10) indicate that the point is 9 units to the right (positive x-direction) and 10 units up (positive y-direction) from the origin (0,0). Similarly, the coordinates for the second point (6, 2) mean moving 6 units right and 2 units up from the origin.

• In essence, coordinates are indicators of a specific point's location on a plane.
• This foundational understanding helps in numerous applications, including calculating slopes, distances, and midpoints.

It's these coordinate points that we utilize to derive meaningful mathematics, like the slope, which quantifies the steepness of the line between these two points.

Linear Equations

A linear equation describes a relationship between two variables, typically x and y, using a simple formula or expression. In its most common form, the equation of a line is expressed as y = mx + b , where m is the slope and b is the y-intercept.
A linear equation graphically represents a straight line in two-dimensional space. The slope indicated by m shows how much y changes with a change in x. In our specific problem, once we determine the slope, we see how this value is integrated into a broader equation.

• The slope, calculated as 8/3, tells us how quickly y increases with x.
• A positive slope indicates that as x increases, y also increases.

Understanding linear equations offers powerful insights into trends and relationships. For example, if the slope is steep, y changes rapidly with x. If it's flat, y changes slowly or not at all.

Algebraic Concepts

Algebraic concepts form the foundation of many mathematical processes and principles. They involve manipulating symbols and letters to solve equations and understand relationships between variables. The slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) is a prime example of using algebra to solve a geometric problem.
The formula embodies the change in y-values (vertical change) over the change in x-values (horizontal change). By substituting the coordinates (9, 10) and (6, 2) into the formula, we determine the slope. Algebraically, this involves:

• Calculating the difference in y, which is 2 - 10 = -8.
• Calculating the difference in x, which is 6 - 9 = -3.
• Simplifying the quotient \( \frac{-8}{-3} \) to \( \frac{8}{3} \).

This simplification is a display of algebraic skill, achieving a clear numerical value that represents the line's slope. Mastery of such algebraic concepts is crucial in exploring more complex mathematical and real-world issues.