Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Find the slope (if defined) of the line that passes through the given points. $$(-4,-3) \text { and }(5,0)$$

Step-by-Step Solution

Answer

The slope of the line is \( \frac{1}{3} \).

Step 1: Recall the Slope Formula

The slope \( m \) of the 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

From the given points, we have the first point \( (x_1, y_1) = (-4, -3) \) and the second point \( (x_2, y_2) = (5, 0) \).

Step 3: Substitute Points into the Slope Formula

Replace \( x_1, y_1, x_2, \) and \( y_2 \) in the slope formula with the corresponding values from the points: \( x_1 = -4, y_1 = -3, x_2 = 5, y_2 = 0 \). So the formula becomes: \[ m = \frac{0 - (-3)}{5 - (-4)} \].

Step 4: Simplify the Expression

Simplify the numerator and denominator of the expression: \[ m = \frac{0 + 3}{5 + 4} = \frac{3}{9} \].

Step 5: Reduce the Fraction

Reduce the fraction \( \frac{3}{9} \) by dividing both numerator and denominator by their greatest common divisor, which is 3. Thus, \[ m = \frac{1}{3} \].

Key Concepts

Slope Formula

The slope formula is essential in coordinate geometry, especially when calculating the steepness or inclination of a line. It is commonly represented as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two distinct points on the line.

This formula calculates how much the line rises or falls vertically (change in \( y \)) for a horizontal distance (change in \( x \)). The slope value tells us whether the line increases, decreases, or is constant.
If the slope is positive, it means the line rises as it moves from left to right. A negative slope indicates a line that falls, while a zero slope means the line is horizontal. An undefined or infinite slope is found in vertical lines, where \( x_2 - x_1 = 0 \).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using the coordinate plane to study geometric shapes and evaluate their properties. One of its most common uses is determining the relationship and position of two or more points.
In the context of finding the slope, we focus on identifying the coordinates of points through which the line passes. For example, when determining the slope of a line through the points \((-4, -3)\) and \((5, 0)\), we recognize these as

• First point: \((x_1, y_1) = (-4, -3)\)
• Second point: \((x_2, y_2) = (5, 0)\)

Placing these points onto the xy-plane allows us to visualize the line and the differences between these coordinates, crucial for applying the slope formula. Coordinate geometry helps to bridge algebraic and geometric views, making abstract concepts more concrete.

Reducing Fractions

Reducing fractions is an essential skill in mathematics, allowing us to represent numbers in their simplest form. This process involves dividing both the numerator and the denominator of a fraction by their greatest common divisor (GCD).
For instance, consider the fraction \(\frac{3}{9}\) obtained when substituting in the slope formula:

• First, find the GCD of 3 and 9, which is 3.
• Then, divide both the numerator and the denominator by this GCD.

Beginning with \(\frac{3}{9}\), we divide both by 3:\[\frac{3}{9} = \frac{3 \div 3}{9 \div 3} = \frac{1}{3}\]The result, \(\frac{1}{3}\), represents the slope in its simplest form. By reducing fractions, we ensure clarity and simplicity in our representation of mathematical solutions, aiding in easier comparison and interpretation.