Problem 16 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \(\left(4, \frac{1}{3}\right)\) and \(\left(6, \frac{2}{3}\right)\)

Step-by-Step Solution

Answer

The slope is approximately 0.17

Step 1: Identify the coordinates

The given points are \(4, \frac{1}{3}\) and \(6, \frac{2}{3}\). Denote these points as \(x_1, y_1\) and \(x_2, y_2\), respectively. So, we have \(x_1 = 4\), \(y_1 = \frac{1}{3}\), \(x_2 = 6\) and \(y_2 = \frac{2}{3}\).

Step 2: Write the slope formula

The slope formula is given by \[m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(m\) represents the slope of the line.

Step 3: Substitute the coordinates into the slope formula

Substitute \(x_1 = 4\), \(y_1 = \frac{1}{3}\), \(x_2 = 6\), and \(y_2 = \frac{2}{3}\) into the slope formula: \[m = \frac{\frac{2}{3} - \frac{1}{3}}{6 - 4} \] Simplify the numerator and the denominator separately.

Step 4: Simplify the expression

Calculate the differences: The numerator is \(\frac{2}{3} - \frac{1}{3} = \frac{1}{3}\) and the denominator is \((6 - 4 = 2)\). Thus, the slope is \[m = \frac{\frac{1}{3}}{2} = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \].

Step 5: Round the slope to the nearest hundredth

Convert \(\frac{1}{6}\) to a decimal and round to the nearest hundredth: \[m \approx 0.17 \].

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols to represent numbers and express mathematical relationships. It helps us simplify and solve equations. When calculating the slope, we use algebraic methods to handle various operations with coordinates and fractions. By using variables like \( x_1 \), \( y_1 \), \( x_2 \), and \( y_2 \), we can generalize the process and apply it to any pair of points.

Coordinates

Coordinates are pairs of numbers that define the position of a point on a plane. In our exercise, the points \( (4, \frac{1}{3}) \) and \( (6, \frac{2}{3}) \) represent specific locations. The first number in each pair is the x-coordinate, and the second is the y-coordinate. They tell us how far along each axis the point is located. By identifying these coordinates, we set the stage for applying the slope formula.

Slope Formula

The slope formula helps determine the steepness of a line connecting two points. It is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \], where \( m \) represents the slope. This formula takes the difference in y-coordinates (rise) and divides it by the difference in x-coordinates (run). By substituting the coordinates of our points into the formula, we find the slope which tells us how much the line ascends or descends as it moves from left to right.

Fractions

Fractions represent parts of a whole and are very common in slope calculations. In our case, we have fractions like \( \frac{1}{3} \) and \( \frac{2}{3} \). When working with the slope formula, we often need to subtract, multiply, or convert fractions. For example, subtracting \( \frac{1}{3} \) from \( \frac{2}{3} \) gives us \( \frac{1}{3} \). Converting this into a decimal helps to present the slope in a more familiar form, especially when rounding to the nearest hundredth: \( \frac{1}{6} \approx 0.17 \).