Problem 47 - A Graphical Approach to College Algebra

Question

Find the slope (if defined) of the line that passes through the given points.{$ \left(\frac{1}{2},-\frac{2}{3}\right) \text { and }\left(-\frac{3}{4}, \frac{1}{6}\right) \end{aligned}

Step-by-Step Solution

Answer

The slope is \(-\frac{2}{3}\).

Step 1: Identify the Coordinates

The two points given are \( \left(\frac{1}{2}, -\frac{2}{3}\right) \) and \( \left(-\frac{3}{4}, \frac{1}{6}\right) \). Identify these coordinates as \((x_1, y_1) = \left(\frac{1}{2}, -\frac{2}{3}\right)\) and \((x_2, y_2) = \left(-\frac{3}{4}, \frac{1}{6}\right)\).

Step 2: Recall the Slope Formula

The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Calculate the Differences in the Coordinates

Calculate the difference in the \(y\)-coordinates: \[ y_2 - y_1 = \frac{1}{6} - \left(-\frac{2}{3}\right) = \frac{1}{6} + \frac{2}{3} \]. Convert \(\frac{2}{3}\) to \(\frac{4}{6}\) to have a common denominator: \[ \frac{1}{6} + \frac{4}{6} = \frac{5}{6} \]. Now, calculate the difference in the \(x\)-coordinates: \[ x_2 - x_1 = -\frac{3}{4} - \frac{1}{2} = -\frac{3}{4} - \frac{2}{4} = -\frac{5}{4} \].

Step 4: Substitute into the Slope Formula

Substitute the differences calculated in Step 3 into the slope formula: \[ m = \frac{\frac{5}{6}}{-\frac{5}{4}} \].

Step 5: Simplify the Expression

Simplify \( \frac{\frac{5}{6}}{-\frac{5}{4}} \): Multiply by the reciprocal of \(-\frac{5}{4}\): \[ m = \frac{5}{6} \times \frac{-4}{5} = -\frac{20}{30} \].Simplify \(-\frac{20}{30}\) to \(-\frac{2}{3}\).

Step 6: Confirm the Slope

Thus, the slope \(m\) of the line passing through the given points is \(-\frac{2}{3}\).

Key Concepts

Slope

Understanding the concept of **slope** is crucial in algebra and coordinate geometry. The slope of a line indicates its steepness and direction. When given two points, the slope is found by using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This formula provides a ratio that represents the vertical change (rise) over the horizontal change (run) between two points on a line.
For example, in the problem, we computed the slope for the points \( \left(\frac{1}{2}, -\frac{2}{3}\right) \) and \( \left(-\frac{3}{4}, \frac{1}{6}\right) \).

• The **vertical change** was \( \frac{5}{6} \), which is the difference between the \( y \)-coordinates.
• The **horizontal change** was \( -\frac{5}{4} \), the difference between the \( x \)-coordinates.

This calculates to a slope of \( -\frac{2}{3} \), indicating a line that decreases in height as it moves from left to right. This negative slope tells us the line goes downward when viewed in a standard graph layout.

Coordinate Geometry

**Coordinate geometry**, or analytic geometry, involves the algebraic representation of geometric figures. When working with lines, we utilize coordinates to define their equations and properties. Each point on a line can be described using an \((x, y)\) coordinate, showing its position on the plane.
In our slope problem, the points \( \left(\frac{1}{2}, -\frac{2}{3}\right) \) and \( \left(-\frac{3}{4}, \frac{1}{6}\right) \) are located on the coordinate plane. Knowing these coordinates allows us not just to find the slope but to also write the line's equation, calculate distances, and examine intersections.

• **Coordinate plane**: A grid where each location is defined by an \((x, y)\) pair.
• **Line**: A set of infinite points that has constant slope between any two points.
• **Equation of a line**: Often expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

This shows how coordinate geometry connects algebra to visual graphs, providing powerful tools for understanding and solving geometric problems.

Fraction Operations

***Fraction operations*** are essential for correctly handling algebraic expressions involving fractions. Fractions represent parts of a whole and are necessary for calculations involving division. Specifically, to manage slopes given as fractions, we follow specific steps:
1. **Finding a common denominator**: Before adding or subtracting fractions, like the \( y \)-coordinate difference \( \frac{1}{6} + \frac{2}{3} \), adjust them to have a common denominator (in this case, 6).2. **Simplifying fractions**: Reduce fractions to their simplest form by finding the greatest common divisor (GCD) of the numerator and denominator. For instance, simplifying \( -\frac{20}{30} \) to \( -\frac{2}{3} \).
3. **Multiplying by reciprocal**: When dividing by a fraction, multiply by the reciprocal to simplify calculations, as shown when calculating the slope \( m = \frac{\frac{5}{6}}{-\frac{5}{4}} \).

• **Addition/Subtraction**: Use a common denominator to perform these operations.
• **Multiplication/Division**: Multiply both numerators and denominators across, or multiply by the inverse.

Mastering these operations allows easier manipulation of equations involving fractions, crucial in coordinate geometry and slope calculations.