Problem 30 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \(\left(\frac{1}{2}, 2\right),\left(\frac{1}{4}, \frac{1}{2}\right)\)

Step-by-Step Solution

Answer

The slope of the line is 6.

Step 1 - Identify the coordinates

Recognize the coordinates of the given points. The first point is \(\frac{1}{2}, 2\) and the second point is \( \frac{1}{4}, \frac{1}{2} \).

Step 2 - Recall the formula for slope

The formula to find the slope \(m\) of a line given two 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 values

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

Step 4 - Simplify the numerator

Calculate the difference in the y-coordinates (the numerator): \[ \frac{1}{2} - 2 = \frac{1}{2} - \frac{4}{2} = -\frac{3}{2} \]

Step 5 - Simplify the denominator

Calculate the difference in the x-coordinates (the denominator): \[ \frac{1}{4} - \frac{1}{2} = \frac{1}{4} - \frac{2}{4} = -\frac{1}{4} \]

Step 6 - Division of fractions

Divide the simplified numerator by the simplified denominator: \[ m = \frac{-\frac{3}{2}}{-\frac{1}{4}} = \frac{-3}{2} \times \frac{4}{-1} = 6 \]

Key Concepts

slope formula

The slope of a line is a measure of its steepness and direction. To find the slope, we use the slope formula, which compares two points on the line. This formula is essential in coordinate geometry. Here's how you can understand it step by step.
The formula for the slope (m) is:
m = \( \frac{y_2 - y_1}{x_2 - x_1} \)
where \( (x_1, y_1) \) and \( (x_2, y_2) \) are any two points on the line. The difference in the y-coordinates \( (y_2 - y_1) \) is divided by the difference in the x-coordinates \( (x_2 - x_1) \). This gives a ratio that represents the slope of the line.
To find the slope between the points \( \left(\frac{1}{2}, 2\right) \) and \( \left(\frac{1}{4}, \frac{1}{2} \right)\), we substitute the coordinates into the formula:
m = \( \frac{\frac{1}{2} - 2}{\frac{1}{4} - \frac{1}{2} } \)
After simplification, we find that the slope of the line is 6.
Understanding the slope formula helps in analyzing various properties of lines in coordinate geometry.

coordinate geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses coordinates to represent geometric figures. This subject combines algebra and geometry to solve problems using a coordinate plane.
In a coordinate plane:

• Every point is represented by an ordered pair \( (x, y) \), which corresponds to its horizontal (x-axis) and vertical (y-axis) position.
• Lines can be examined using their slopes, intercepts, and equations.

When given two points, like \( \left(\frac{1}{2}, 2\right) \) and \( \left(\frac{1}{4}, \frac{1}{2} \right) \), we can analyze them using their coordinates. These coordinates are critical in applying the slope formula.
Coordinate geometry is powerful because it brings together visual interpretations and algebraic representations. This makes it easier for students to solve complex geometric problems.

division of fractions

Dividing fractions is a crucial skill in mathematics, especially when simplifying expressions like the slope formula. Here's a quick review on how to handle fraction division.
When dividing fractions, you multiply by the reciprocal of the divisor. For example, to divide \(-\frac{3}{2} \) by \(-\frac{1}{4} \):
\( -\frac{3}{2} \div -\frac{1}{4} = -\frac{3}{2} \times -4 \)
Multiplication of the fractions is straightforward: multiply the numerators and then the denominators:
\( -\frac{3}{2} \times -4 = \frac{12}{2} = 6 \)
This gives a simplified result of 6. In our slope problem, these steps helped us conclude that the slope of the line is 6.
Mastering division of fractions is essential for various mathematical problems, including finding the slope in coordinate geometry.