Problem 7 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

The slope is 2.

Step 1 - Understand the Slope Formula

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

Step 2 - Identify Coordinates

Identify the coordinates \(x_1, y_1\) and \(x_2, y_2\) from the given points. Here, \(x_1 = \frac{3}{8}\), \(y_1 = -\frac{1}{2}\), \(x_2 = -\frac{5}{8}\), and \(y_2 = -\frac{5}{2}\).

Step 3 - Subtract y-coordinates

Subtract the y-coordinates: \( y_2 - y_1 = -\frac{5}{2} - \left( -\frac{1}{2} \right) = -\frac{5}{2} + \frac{1}{2} = -\frac{4}{2} = -2 \).

Step 4 - Subtract x-coordinates

Subtract the x-coordinates: \( x_2 - x_1 = -\frac{5}{8} - \left( \frac{3}{8} \right) = -\frac{5}{8} - \frac{3}{8} = -\frac{8}{8} = -1 \).

Step 5 - Divide and Simplify

Divide the results from steps 3 and 4 to find the slope: \( m = \frac{-2}{-1} = 2 \). Hence, the slope of the line is 2.

Key Concepts

slope formula

To find the slope of a line, you need to understand and use the slope formula. The slope formula helps you determine the steepness or incline of a line that passes through two points. The formula is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this formula, \( m \) represents the slope, and \( x_1, y_1 \) and \( x_2, y_2 \) are the coordinates of the two points through which the line passes. By applying this formula, you can easily calculate the slope. This formula is fundamental in coordinate geometry as it helps describe how one variable changes in relation to another.

coordinate geometry

Coordinate geometry, also known as analytic geometry, allows us to represent geometric figures and analyze their properties using a coordinate system. Here, we work with the Cartesian plane which is defined by the x-axis and y-axis. Every point on this plane can be represented by a pair of coordinates, \( (x, y) \). To solve the slope exercise, we use the coordinates of the given points. In our example, the coordinates are:

• \( \frac{3}{8}, -\frac{1}{2} \) and \( -\frac{5}{8}, -\frac{5}{2} \)

By using these coordinates in our calculations, we follow a systematic way of solving geometric problems.

subtraction of fractions

When dealing with the slope formula, you'll often need to subtract fractions. For example, to find the change in y-coordinates, we perform the subtraction:

• \( y_2 - y_1 = -\frac{5}{2} - (-\frac{1}{2}) = -\frac{5}{2} + \frac{1}{2} \)

Here we use the rule that subtracting a negative fraction is the same as adding its positive counterpart. The steps are as follows:

• First, change the subtraction to addition where necessary.
• Find a common denominator if needed.
• Perform the operation on both the numerators.

In our case, \( -\frac{5}{2} + \frac{1}{2} = -\frac{4}{2} = -2 \). This operation is crucial for finding the correct numerator in the slope formula.

simplifying fractions

Simplifying fractions is an essential skill when working with the slope formula. Once you subtract the coordinates, you may need to simplify the resulting fraction. For example, let's look at the calculation of the change in x-coordinates:

• \( x_2 - x_1 = -\frac{5}{8} - \frac{3}{8} \)
• \( -\frac{5}{8} - \frac{3}{8} = -\frac{8}{8} = -1 \)

The fraction \( -\frac{8}{8} \) simplifies to -1. Simplifying fractions involves reducing them to their simplest form by dividing both the numerator and the denominator by their greatest common divisor. After simplifying, the final step is:

• \( m = \frac{-2}{-1} \)

Simplifying \( \frac{-2}{-1} \) gives us the slope of 2. Always double-check your steps to ensure accuracy.