Problem 86 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (-3,2),(5,6) $$

Step-by-Step Solution

Answer

The slope is \( \frac{1}{2} \).

Step 1: Recall the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula expresses the change in \(y\) over the change in \(x\).

Step 2: Identify the Coordinates

From the given points \((-3, 2)\) and \((5, 6)\), identify the coordinates as follows: \((x_1, y_1) = (-3, 2)\) and \((x_2, y_2) = (5, 6)\).

Step 3: Substitute the Coordinates into the Formula

Substitute \(x_1 = -3\), \(y_1 = 2\), \(x_2 = 5\), and \(y_2 = 6\) into the slope formula: \( m = \frac{6 - 2}{5 - (-3)} \).

Step 4: Simplify the Expression

Calculate the expression \( m = \frac{6 - 2}{5 + 3} = \frac{4}{8} \). Simplify \( \frac{4}{8} \) to get \( \frac{1}{2} \).

Step 5: Finalize the Slope Value

The slope of the line that passes through the points \((-3, 2)\) and \((5, 6)\) is \( \frac{1}{2} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where points on a plane are described using an ordered pair of numbers. These numbers are known as coordinates. Each point on a coordinate plane is defined by its position along the x-axis (horizontal) and y-axis (vertical). For example, in the problem at hand, we have two points:

• Point A:
  • Coordinates: (-3, 2)
  • Position: 3 units to the left and 2 units up from the origin
• Point B:
  • Coordinates: (5, 6)
  • Position: 5 units to the right and 6 units up from the origin

Coordinate geometry is essential for understanding the basic properties of shapes and positions in the plane. By using coordinate points, mathematicians can calculate lengths, midpoints, slopes, and more with precision.

Slope Formula

The slope formula is a fundamental concept in coordinate geometry used to calculate the steepness of a line. It is commonly represented by the letter 'm' and calculated using the coordinates of two distinct points on the line. The formula for slope is: \[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula provides the 'rise over run,' which is how much the line goes up or down as it moves from left to right. For the points given in the exercise, (-3, 2) and (5, 6):

• Rise (difference in y-coordinates): 6 - 2 = 4
• Run (difference in x-coordinates): 5 - (-3) = 5 + 3 = 8

Using these values in our formula, the slope, \(m\), is calculated as \( \frac{4}{8} \). The slope is a measure of how quickly or slowly a line ascends or descends. In this context, a positive slope indicates the line rises to the right, while a negative slope would mean it descends.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. This means expressing a fraction in such a way that the numerator and the denominator have no common factors other than 1. When we calculated the slope as \( \frac{4}{8} \), we went on to simplify it.
The fraction \( \frac{4}{8} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4 in this case. Performing the division gives:

• Numerator: \( 4 \div 4 = 1 \)
• Denominator: \( 8 \div 4 = 2 \)

So, \( \frac{4}{8} \) becomes \( \frac{1}{2} \).This process is crucial, not only to simplify results in mathematics but also to ensure accuracy and clarity when conveying mathematical computations or results.