Problem 5 - Elementary Algebra

Question

For exercises 1-8, find the slope of the line that passes through the given points. $$ \left(\frac{3}{8}, \frac{2}{5}\right)\left(\frac{9}{16}, \frac{3}{4}\right) $$

Step-by-Step Solution

Answer

The slope is \frac{28}{15}.

Step 1: Identify the given points

The given points are \ \( \left( \frac{3}{8}, \frac{2}{5} \right) \) and \ \( \left( \frac{9}{16}, \frac{3}{4} \right) \). Label these points as \ \( \left( x_1, y_1 \right) \) and \ \( \left( x_2, y_2 \right) \) respectively. Hence, \ \( x_1 = \frac{3}{8}, y_1 = \frac{2}{5}, x_2 = \frac{9}{16}, y_2 = \frac{3}{4} \).

Step 2: Write the formula for the slope of a line

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

Step 3: Substitute the given values into the slope formula

Using the values from Step 1, the slope formula becomes \ \[ m = \frac{\frac{3}{4} - \frac{2}{5}}{\frac{9}{16} - \frac{3}{8}} \].

Step 4: Simplify the numerator

Calculate \ \( y_2 - y_1 \) as follows: \ \[ y_2 - y_1 = \frac{3}{4} - \frac{2}{5} = \frac{15}{20} - \frac{8}{20} = \frac{7}{20} \].

Step 5: Simplify the denominator

Calculate \ \( x_2 - x_1 \) as follows: \ \[ x_2 - x_1 = \frac{9}{16} - \frac{3}{8} = \frac{9}{16} - \frac{6}{16} = \frac{3}{16} \].

Step 6: Divide the simplified numerator by the simplified denominator

Finally, calculate the slope \ \( m \) using the simplified terms: \ \[ m = \frac{\frac{7}{20}}{\frac{3}{16}} = \frac{7}{20} \times \frac{16}{3} = \frac{7 \times 16}{20 \times 3} = \frac{112}{60} = \frac{28}{15} \].

Key Concepts

slope formula

The slope of a line measures its steepness or inclination. To find the slope, we use a special formula. This is called the slope formula. The slope formula is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two distinct points on the line.
In simple terms, the formula calculates the difference in y-values (rise) divided by the difference in x-values (run). The result is a number that tells us how steep the line is. For example, if the slope is 2, this means for every unit we move to the right, the line goes up by 2 units.

points on a coordinate plane

Points on a coordinate plane are locations defined by an x-coordinate and a y-coordinate. For instance, the point \(\left(\frac{3}{8}, \frac{2}{5}\right)\) tells us the location is \(\frac{3}{8}\) units across (to the right) from the origin and \(\frac{2}{5}\) units up.
The coordinates are given in the format \((x, y)\). In this exercise, the points are fractions, making it slightly more challenging. Using these points, we can calculate the slope of the line that passes through them.

simplifying fractions

Simplifying fractions is making a fraction as simple as possible. This can involve reducing it to its smallest form. For example, \(\frac{7}{20}\) is already in simplest form. But after calculating the slope, we got \(\frac{112}{60}\), which simplifies to \(\frac{28}{15}\).
To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). This can make calculations much easier.

numerator and denominator in slope calculation

In the slope formula, the numerator is the difference between the y-values. The denominator is the difference between the x-values. For the points \(\left(\frac{3}{8}, \frac{2}{5}\right)\) and \(\left(\frac{9}{16}, \frac{3}{4}\right)\), the calculations are as follows:

• Numerator: \(y_2 - y_1 = \frac{3}{4} - \frac{2}{5} = \frac{7}{20}\)
• Denominator: \(x_2 - x_1 = \frac{9}{16} - \frac{3}{8} = \frac{3}{16}\)

The final step is dividing the numerator by the denominator to get the slope \[ m = \frac{7}{20} \div \frac{3}{16} = \frac{28}{15} \]. The fractions here make it slightly more complex, but careful calculation gives you the correct slope.