Problem 5 - Contemporary Precalculus

Question

Find the slope of the line through the given points. $$(1 / 4,0) ;(3 / 4,2)$$

Step-by-Step Solution

Answer

Answer: The slope of the line passing through the points (1/4, 0) and (3/4, 2) is 4.

Step 1: Identify the coordinates

Assign the coordinates of the two given points to \((x_1, y_1)\) and \((x_2, y_2)\). In this case: $$x_1 = \frac{1}{4},\, y_1 = 0, \, x_2 = \frac{3}{4},\, y_2 = 2$$

Step 2: Apply the slope formula

Insert the coordinates from Step 1 into the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 0}{\frac{3}{4} - \frac{1}{4}}$$

Step 3: Simplify the expression

Simplify the fraction in the denominator and numerator: $$m = \frac{2}{\frac{2}{4}}$$

Step 4: Compute the slope

Now, compute the slope \(m\) by multiplying the numerator by the reciprocal of the denominator: $$m = 2 \times \frac{4}{2} = 2 \times 2$$ Then we get the final result: $$m=4$$ So the slope of the line passing through the points \((1/4,0)\) and \((3/4,2)\) is \(4\).

Key Concepts

Coordinate Geometry

Coordinate geometry, often called analytic geometry, is a fascinating branch of mathematics. It connects algebra with geometry and allows us to use coordinates to describe geometric figures. Each point in a coordinate plane is represented by an \(x\)-coordinate and a \(y\)-coordinate. These coordinates point us to the exact position in a two-dimensional space.

For instance, the points \( (1/4, 0) \) and \( (3/4, 2) \) mean that you start your journey at the origin (0,0) and move to the right by \(1/4\) and \(3/4\) units respectively along the X-axis. After this move, you then follow the Y-axis by \(0\) and \(2\) units respectively.

Understanding coordinate geometry provides us with a powerful tool to compute not just positions, but also relationships between the points such as distance and slope.

Slope Formula

The slope of a line is an essential concept when working with coordinate geometry. It's a measure of how steep a line is. In other words, it tells us how much the line rises or falls as we move along it. The slope is an incredibly useful tool when trying to understand the relationships between points on a line.

The formula to calculate the slope (\(m\)) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

In simple terms, the formula is a ratio of the difference in \(y \) coordinates (vertical change) to the difference in \(x \) coordinates (horizontal change).

By using this formula with the points \( (1/4, 0) \) and \( (3/4, 2) \), we identified the slope to be 4. This means for every one unit you move horizontally from left to right, the line rises by 4 units.

Fractions in Algebra

Fractions are numbers that represent parts of a whole and are crucial in algebra. When dealing with fractions in algebra, especially in the context of coordinate geometry and slope calculations, some additional attention is required.

To find the slope between two points, you often need to manipulate fractions. For instance, in our exercise, we had fractions as coordinates \(\frac{1}{4}\) and \(\frac{3}{4}\).
Performing operations like subtraction or division with fractions requires you to remember:

• The common denominator rule for addition and subtraction.
• The reciprocal rule for division, where dividing by a fraction is equivalent to multiplying by its reciprocal.

These concepts enabled us to simplify the expression \(\frac{2}{\frac{2}{4}}\) to give us a slope of 4.
Practicing using fractions in algebra helps in solving geometric problems with more accuracy and ease.