Problem 90 - Algebra A Combined Function

Question

Find the slope of the line through the given points. $$ (2.3,0.2) \text { and }(7.9,5.1) $$

Step-by-Step Solution

Answer

The slope of the line is approximately 0.875.

Step 1: Identify the Points

The given points are \((x_1, y_1) = (2.3, 0.2)\) and \((x_2, y_2) = (7.9, 5.1)\).

Step 2: Recall the Slope Formula

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Substitute the Values into the Formula

Substitute \(x_1 = 2.3\), \(y_1 = 0.2\), \(x_2 = 7.9\), and \(y_2 = 5.1\) into the slope formula: \[ m = \frac{5.1 - 0.2}{7.9 - 2.3} \].

Step 4: Calculate the Difference in Y-Coordinates

Calculate \(5.1 - 0.2 = 4.9\).

Step 5: Calculate the Difference in X-Coordinates

Calculate \(7.9 - 2.3 = 5.6\).

Step 6: Divide the Differences to Find the Slope

Divide the difference in y-coordinates by the difference in x-coordinates: \[ m = \frac{4.9}{5.6} \approx 0.875 \].

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows us to use algebraic equations to describe geometric shapes and analyze their properties. In a two-dimensional coordinate system, each point is represented by a pair of numbers, typically written as \((x, y)\). The two numbers represent the point's horizontal and vertical positions on a graph.

A line in coordinate geometry is often expressed as a set of points that satisfy a linear equation, such as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Understanding the position and nature of a line in a coordinate plane is essential in solving many geometry-related problems.

Coordinate geometry gives us a powerful toolkit to calculate distances, midpoints, and slopes. Whenever you encounter a problem involving points, lines, or curves on a plane, coordinate geometry provides a structured way to solve it by converting geometric relationships into algebraic equations.

Slope Formula

The slope of a line is a measure of its steepness and direction. Calculated as the ratio of the change in the vertical direction to the change in the horizontal direction, the slope tells us how much a line rises or falls as it moves from left to right.

The slope formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on a line. This formula derives from the fact that slope is essentially the "rate of change" between the two points.

Understanding this formula helps you determine whether a line is ascending, descending, or horizontal:

• A positive slope means the line rises as it moves right.
• A negative slope indicates the line falls as it moves right.
• A slope of zero means the line is horizontal.

This formula is fundamental to coordinate geometry, as it allows you to calculate the angle and direction of lines in a plane with ease.

Calculation of Slope

In order to calculate the slope of a line through two points, it is essential to follow a systematic approach using the slope formula. Let's break this process into clear steps:

1. **Identify the Points:** Ensure you have two points, say \((x_1, y_1)\) and \((x_2, y_2)\). For instance, consider \((2.3, 0.2)\) and \((7.9, 5.1)\).
2. **Substitute Values in the Formula:** Use the slope formula by plugging in the values of the coordinates: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5.1 - 0.2}{7.9 - 2.3} \].
3. **Calculate the Differences:** First, find the change in the y-coordinates, \(5.1 - 0.2 = 4.9\). Next, find the change in the x-coordinates, \(7.9 - 2.3 = 5.6\).
4. **Compute the Slope:** Finally, divide the difference in y-values by the difference in x-values: \[ m = \frac{4.9}{5.6} \approx 0.875 \].
Through these straightforward steps, you can determine the precise slope of any line given two points, making it invaluable for tasks ranging from simple graphing to complex analytical problem-solving in coordinate geometry.