Problem 17 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \(\left(-\frac{1}{3}, \frac{1}{5}\right)\) and \(\left(\frac{3}{2}, \frac{1}{4}\right)\)

Step-by-Step Solution

Answer

The slope is approximately 0.03.

Step 1: Identify the coordinates

The given points are \(\big(-\frac{1}{3}, \frac{1}{5}\big)\) and \(\big(\frac{3}{2}, \frac{1}{4}\big)\). Let's denote these coordinates as \(x_1, y_1\) and \(x_2, y_2\) where \(x_1 = -\frac{1}{3}\), \(y_1 = \frac{1}{5}\), \(x_2 = \frac{3}{2}\), and \(y_2 = \frac{1}{4}\).

Step 2: Use the slope formula

The formula for the slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\).

Step 3: Substitute the coordinates into the slope formula

Substitute \((x_1, y_1) = (-\frac{1}{3}, \frac{1}{5})\) and \((x_2, y_2) = (\frac{3}{2}, \frac{1}{4})\) into the slope formula:\[m = \frac{\frac{1}{4} - \frac{1}{5}}{\frac{3}{2} - (-\frac{1}{3})}\]

Step 4: Simplify the numerator

Find the difference between the y-coordinates: \[ \frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \]

Step 5: Simplify the denominator

Find the difference between the x-coordinates: \[ \frac{3}{2} - (-\frac{1}{3}) = \frac{3}{2} + \frac{1}{3} = \frac{9}{6} + \frac{2}{6} = \frac{11}{6} \]

Step 6: Divide the numerator by the denominator

Divide \(\frac{1}{20} \) by \(\frac{11}{6}\): \[ m = \frac{\frac{1}{20}}{\frac{11}{6}} = \frac{1}{20} \cdot \frac{6}{11} = \frac{6}{220} = \frac{3}{110} = 0.027 \]

Key Concepts

coordinate geometry

Coordinate geometry combines algebra and geometry to describe the positions of points on a plane. A point is represented by an ordered pair \((x, y)\), where \(x\) and \(y\) are its coordinates.
In the given exercise, you have two points: \((-\frac{1}{3}, \frac{1}{5})\) and \((\frac{3}{2}, \frac{1}{4})\).
 
Each coordinate pair describes a location on the Cartesian plane. The first number in each pair \(x\) represents the horizontal position, and the second \(y\) represents the vertical position.

The coordinates help you visualize the line passing through these points.

slope formula

The slope of a line measures its steepness and direction. It's calculated as the ratio of the 'rise' (vertical change) to the 'run' (horizontal change) between two points on the line.
The formula for the slope \(m\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
 
Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the two points. In the exercise, you'll substitute the values:
\[ x_1 = -\frac{1}{3} \, \ y_1 = \frac{1}{5} \, \ x_2 = \frac{3}{2} \, \ y_2 = \frac{1}{4} \]
Plugging these into the formula gives:
\[ m = \frac{\frac{1}{4} - \frac{1}{5}}{\frac{3}{2} - (-\frac{1}{3})} \]

fraction simplification

Simplifying fractions means reducing them to their simplest form.
In the exercise, you perform several steps to simplify fractions:

• First, simplify the numerator \- the vertical difference between \(y_2\) and \(y_1\):
  \[ \frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20} \]


• Next, simplify the denominator \- the horizontal difference between \(x_2\) and \(x_1\):
  \[ \frac{3}{2} - (-\frac{1}{3}) = \frac{3}{2} + \frac{1}{3} = \frac{9}{6} + \frac{2}{6} = \frac{11}{6} \]


• Lastly, divide the simplified numerator by the simplified denominator:
  \[ m = \frac{\frac{1}{20}}{\frac{11}{6}} = \frac{1}{20} \cdot \frac{6}{11} = \frac{6}{220} = \frac{3}{110} \]

After rounding, the slope \(m\) is approximately 0.027.

These steps ensure the fractions are easier to compute and interpret.