Problem 39 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ \left(\frac{1}{2}, \frac{2}{3}\right),\left(\frac{5}{6}, \frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The slope of the line is 5.

Step 1: Understanding the Slope Formula

The slope of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1) = \left(\frac{1}{2}, \frac{2}{3}\right)\) and \((x_2, y_2) = \left(\frac{5}{6}, \frac{1}{4}\right)\).

Step 2: Substitute the Points into the Formula

We need to substitute the coordinates into the slope formula: \(m = \frac{\frac{1}{4} - \frac{2}{3}}{\frac{5}{6} - \frac{1}{2}}\).

Step 3: Calculate the Differences

First, calculate the difference between the y-coordinates: \(\frac{1}{4} - \frac{2}{3} = \frac{1 \times 3 - 4 \times 2}{12} = \frac{3 - 8}{12} = \frac{-5}{12}\). Then, compute the difference between the x-coordinates: \(\frac{5}{6} - \frac{1}{2} = \frac{5 \times 1 - 6 \times 1}{12} = \frac{5 - 6}{12} = \frac{-1}{12}\).

Step 4: Compute the Slope

Divide the difference in y-coordinates by the difference in x-coordinates: \(m = \frac{\frac{-5}{12}}{\frac{-1}{12}} = \frac{-5}{12} \times \frac{12}{-1} = 5\).

Step 5: Final Result

After simplifying, the slope of the line passing through the points \(\left(\frac{1}{2}, \frac{2}{3}\right)\) and \(\left(\frac{5}{6}, \frac{1}{4}\right)\) is \(5\).

Key Concepts

Slope Formula

The slope formula is a fundamental tool in understanding the properties of a line. It is given by the equation:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope of the line, and \((x_1, y_1)\) and \((x_2, y_2)\) are two distinct points on the line.
Understanding the slope is crucial because it tells us how steep the line is and the direction in which it slopes. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.
When using the slope formula, always substitute the correct coordinates into \(x_1, y_1\) and \(x_2, y_2\). Ensure that you subtract the corresponding coordinates correctly to avoid errors in your calculations.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometric figures using a coordinate system. In coordinate geometry, each point is uniquely represented by an ordered pair \((x, y)\). These coordinates help us understand the positioning and properties of lines and shapes on a plane.
This branch of mathematics is incredibly helpful for solving real-world problems, where precise positioning and dimensions are necessary. It enables us to calculate distances, midpoints, and slopes, allowing a deeper analysis of shapes and figures.
When dealing with problems like finding the slope, coordinate geometry provides a visual framework that can make understanding algebraic concepts easier. The position of points on the Cartesian plane directly influences the line's slope, making it a crucial aspect of many mathematical problems.

Algebraic Fractions

Algebraic fractions consist of a numerator and a denominator, both of which can be algebraic expressions. They require a firm understanding of fraction operations, such as addition, subtraction, multiplication, and division, as well as simplification methods.
In problems like slope calculation, algebraic fractions appear when points are represented by fractional coordinates. To handle these successfully:

• Ensure you have a common denominator when subtracting fractions, as seen in the subtraction steps for both \(x\)- and \(y\)-coordinates.
• Simplify the resulting fractions by canceling out common factors to make computations more straightforward.
• Always check your final result, since fractions could often be simplified further, leading to a neater and more understandable solution.

By mastering algebraic fractions, students can simplify complex algebraic tasks, including those found in coordinate geometry, with greater ease and confidence.