Problem 52 - Elementary Algebra

Question

Find the slope, if it exists, of the line through the given pairs of points. $$ (-3,-2), \quad(-4,-5) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 3.

Step 1: Identify the given points

The given pairs of points are \((-3, -2)\) and \((-4, -5)\). Let's label them as: $$ (x_1, y_1) = (-3, -2) \\ (x_2, y_2) = (-4, -5) $$

Step 2: Use the slope formula

Use the slope formula to find the slope, m, of the line through the given pairs of points: $$ m = \frac{y2 - y1}{x2 - x1} $$

Step 3: Substitute the given points into the formula

Substitute the coordinates of the given points into the formula: $$ m = \frac{-5 - (-2)}{-4 - (-3)} $$

Step 4: Simplify the expression

Simplify the expression to calculate the slope: $$ m = \frac{-5 + 2}{-4 + 3} \\ m = \frac{-3}{-1} $$

Step 5: Find the slope

Divide the numbers to get the slope: $$ m = 3 $$ The slope of the line through the given pairs of points is 3.

Key Concepts

Slope Formula

Understanding the slope formula is crucial for anyone studying algebra and geometry, as it encapsulates the rate at which one variable changes with respect to another. When you're faced with finding the slope of a line that passes through two points, you'll use the slope formula:
\[\[\begin{align*}\( m = \frac{y_2 - y_1}{x_2 - x_1} \)\end{align*}\]\]
This formula derives from the idea that slope equals the rise over the run, which geometrically signifies the vertical change (the rise) divided by the horizontal change (the run) between two points on a line.
To use the slope formula effectively, perform these algebraic operations:

• Identify and label your points with coordinates \( (x_1, y_1) \and \( (x_2, y_2) \)\).
• Subtract the y-coordinates \( (y_2 - y_1) \) to find the rise.
• Subtract the x-coordinates \( (x_2 - x_1) \) to determine the run.
• Divide the rise by the run to calculate the slope \( (m) \) of the line.

If your computation results in a positive number, the line slopes upward; conversely, a negative slope indicates that the line slopes downward. In our example, after applying the formula, we obtained a slope \( (m = 3) \), which suggests a positive rate of change and consequently an upward-sloping line.

Coordinate Pairs

Coordinate pairs are the foundation of the coordinate plane system, acting as the addresses for locating points on a grid. Each pair is composed of an x-coordinate and a y-coordinate, usually written in the form \( (x, y) \). The x-coordinate specifies the position along the horizontal axis, while the y-coordinate indicates the position along the vertical axis.
For example, in our exercise, we had the coordinate pairs \( (-3, -2) \) and \( (-4, -5) \). Here's how to interpret these pairs:

• The first number, the x-coordinate, tells us how far to move left or right from the origin (the center of the grid where the x-axis and y-axis intersect). Negative values move to the left, while positive values take us to the right.
• The second number, the y-coordinate, tells us how far to move up or down from the origin. Negative values indicate a downward movement, and positive values indicate an upward one.

When plotting these on a graph, you would start at the origin, move along the x-axis according to the x-coordinate, and then move parallel to the y-axis according to the y-coordinate. In algebra, we frequently work with coordinate pairs to solve for the slope, distance, or midpoint between points.

Algebraic Operations

Algebraic operations are the bread and butter of solving mathematical problems involving equations and formulas. When finding the slope from points, you will commonly engage in basic operations such as subtraction and division. These operations are used to compute the rise and the run between two points on a line.
In our textbook solution, we executed subtraction to find the difference in the y-coordinates \( (-5 - (-2)) \) and the x-coordinates \( (-4 - (-3)) \). The operation involving subtraction of negative numbers often confuses students, but remember that subtracting a negative is the same as adding a positive.
We then divided the differences to find the slope, completing another essential algebraic operation. It is important to simplify each expression fully and carry out operations in the correct order following the rules of arithmetic. In the given example, recognizing that \( (-3)/(-1) \) results in a positive 3 illustrates how understanding algebraic manipulations can help us interpret the direction and steepness of a line.