Problem 3 - Elementary Algebra

Question

For exercises 1-8, find the slope of the line that passes through the given points. $$ \left(\frac{1}{9}, \frac{2}{15}\right)\left(\frac{5}{9}, \frac{11}{15}\right) $$

Step-by-Step Solution

Answer

The slope is \( \frac{27}{20} \).

Step 1: Identify the given points

The given points are \( \left( \frac{1}{9}, \frac{2}{15} \right) \) and \( \left( \frac{5}{9}, \frac{11}{15} \right) \).

Step 2: Recall the slope formula

The formula for the slope (m) of the line that passes 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: Substitute the coordinates into the slope formula

Substitute the coordinates \( x_1 = \frac{1}{9}, y_1 = \frac{2}{15}, x_2 = \frac{5}{9}, \) and \( y_2 = \frac{11}{15} \) into the slope formula. \[ m = \frac{\frac{11}{15} - \frac{2}{15}}{\frac{5}{9} - \frac{1}{9}} \]

Step 4: Simplify the numerator

Simplify the numerator \( \frac{11}{15} - \frac{2}{15} \): \[ \frac{11}{15} - \frac{2}{15} = \frac{11 - 2}{15} = \frac{9}{15} \]

Step 5: Simplify the denominator

Simplify the denominator \( \frac{5}{9} - \frac{1}{9} \): \[ \frac{5}{9} - \frac{1}{9} = \frac{5 - 1}{9} = \frac{4}{9} \]

Step 6: Divide the simplified numerator by the simplified denominator

Divide \( \frac{9}{15} \) by \( \frac{4}{9} \): \[ m = \frac{\frac{9}{15}}{\frac{4}{9}} = \frac{9}{15} \times \frac{9}{4} = \frac{81}{60} = \frac{27}{20} \]

Key Concepts

slope formula

The slope formula is vital for calculating how steep a line is in coordinate geometry. The formula for the slope, represented as 'm', is derived from the change in y-coordinates divided by the change in x-coordinates.
The slope formula is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
You can use it when you know two points on a line, where one point is \( (x_1, y_1) \) and the other is \( (x_2, y_2) \).
To apply the formula:

• First, identify your points ((x1, y1) & (x2, y2))
• Next, find the differences: (y2 - y1) and (x2 - x1)
• Finally, divide the difference in y-coordinates by the difference in x-coordinates

This method helps maintain consistency and ensures accuracy.
In our example, the points are \(\left( \frac{1}{9}, \frac{2}{15} \right)\) and \(\left( \frac{5}{9}, \frac{11}{15} \right)\).
Plug these values into the slope formula to get:
\[ m = \frac{\frac{11}{15} - \frac{2}{15}}{\frac{5}{9} - \frac{1}{9}}. \]

coordinate geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry. It's about understanding geometric shapes using a coordinate system.
Here, the position is represented by coordinates on the x and y axes. The axes intersect at the origin (0, 0).
Any point in this system is represented by (x, y) coordinates. These are often called ordered pairs.
For example, \(\left( \frac{1}{9}, \frac{2}{15} \right)\) means 1/9 units along the x-axis and 2/15 units along the y-axis.
Key items in coordinate geometry are:

• Points (defined by coordinates)
• Lines (defined by points or equations)
• Slopes (show the steepness of a line)

To find the slope between two points in coordinate geometry, we utilize their coordinates as shown in the slope formula.

fractions

Fractions represent parts of a whole and are crucial in many mathematical calculations. A fraction is given by \(\frac{a}{b}\) where 'a' is the numerator and 'b' is the denominator.
In our slope problem, we combined and simplified fractions multiple times.
Key operations include:

• Adding/Subtracting Fractions: Ensure you have a common denominator before adding or subtracting the numerators.
  For example, \(\frac{11}{15} - \frac{2}{15} = \frac{11 - 2}{15} = \frac{9}{15}\).
• Multiplying Fractions: Multiply the numerators and denominators separately.
  For instance, \(\frac{9}{15} \times \frac{9}{4} = \frac{81}{60}\).
• Simplifying Fractions: Find common factors to reduce the fraction.
  Simplifying \(\frac{81}{60}\) gives you \(\frac{27}{20}\).

These steps are essential as they simplify complex fraction problems.