Problem 27 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((0,3),(5,0)\)

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{5}\).

Step 1: Write down the formula for the slope

The slope (m) 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 the coordinates

Identify the coordinates of the given points. Here, \( (x_1, y_1) = (0, 3) \) \( (x_2, y_2) = (5, 0) \)

Step 3: Substitute the coordinates into the formula

Substitute the identified coordinates into the slope formula: \[ m = \frac{0 - 3}{5 - 0} \]

Step 4: Simplify the fraction

Simplify the expression: \[ m = \frac{-3}{5} \]

Key Concepts

slope formula

Understanding the slope formula is crucial in coordinate geometry. The slope of a line indicates its steepness and direction. It's represented by the letter \( m \). The formula to find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is written as:

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

To use this formula, follow these steps:

• Label your points. The first point is \( (x_1, y_1) \) and the second point is \( (x_2, y_2) \).
• Subtract the y-coordinates \( (y_2 - y_1) \) to find the change in y. This represents the vertical change.
• Subtract the x-coordinates \( (x_2 - x_1) \) to find the change in x. This represents the horizontal change.
• Divide the difference in y by the difference in x to find the slope \( m \).

For example, if you have points \( (0, 3) \) and \( (5, 0) \), follow these steps:

• Label the coordinates: \( (x_1, y_1) = (0, 3) \) and \( (x_2, y_2) = (5, 0) \).
• Find the change in y: \( 0 - 3 = -3 \).
• Find the change in x: \( 5 - 0 = 5 \).
• Divide: \( \frac{-3}{5} \).

The slope of the line is \( \frac{-3}{5} \).

coordinate geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry through graphs and coordinates. It allows us to describe geometric shapes and their properties using algebraic equations.

In coordinate geometry, a plane is defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Points on the plane are represented by coordinates \( (x, y) \). To understand how lines work in this plane, the slope formula is very useful.

• Coordinates: Each point on the plane is defined by an x-coordinate (horizontal position) and a y-coordinate (vertical position).
• Distance Between Points: The formula for the distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
• Midpoint: The midpoint \( M \) of a line segment between \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Using coordinate geometry makes it easier to solve problems involving distances, midpoints, slopes, and the equations of lines. For example, in our exercise, we use the points \( (0, 3) \) and \( (5, 0) \) to determine the slope using the slope formula.

simplifying fractions

Simplifying fractions is an essential skill in algebra, which makes calculations easier and results more readable. A fraction is simplified when the numerator (top number) and the denominator (bottom number) have no common factors other than 1.

Here are the steps to simplify a fraction:

• Find the Greatest Common Divisor (GCD): Identify the largest number that divides both the numerator and the denominator without a remainder.
• Divide Both Terms by the GCD: Simplify the fraction by dividing the numerator and the denominator by their GCD.

For the fraction \( \frac{-3}{5} \) in our example, the GCD is 1 because 3 and 5 are prime numbers and have no common factors other than 1. Thus, the fraction is already in its simplest form.

Let's look at a different example:

• Consider the fraction \( \frac{8}{12} \).
• The GCD of 8 and 12 is 4.
• Divide both the numerator and the denominator by 4: \( \frac{8 \div4}{12 \div4} = \frac{2}{3} \).

Now, \( \frac{2}{3} \) is the simplified form of \( \frac{8}{12} \). Simplifying fractions helps us work with more straightforward and manageable numbers, which is very helpful when solving algebraic problems.