Problem 2 - Elementary Algebra

Question

For exercises 1-8, find the slope of the line that passes through the given points. $$ (8,14)(15,42) $$

Step-by-Step Solution

Answer

The slope of the line is 4.

Step 1 - Identify the Given Points

The given points are \(8, 14\) and \(15, 42\). Let \(x_1, y_1\) be \(8, 14\) and \(x_2, y_2\) be \(15, 42\).

Step 2 - Recall the Slope Formula

The formula for the slope, \(m\), of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3 - Substitute the Values into the Slope Formula

Substitute \(x_1 = 8, y_1 = 14, x_2 = 15, y_2 = 42\) into the formula: \[ m = \frac{42 - 14}{15 - 8} \]

Step 4 - Simplify the Expression

Simplify the numerator and the denominator: \[ m = \frac{28}{7} \]

Step 5 - Compute the Slope

Divide the numerator by the denominator to find the slope: \[ m = 4 \]

Key Concepts

Finding the Slope of a Line

The slope of a line is a measure of its steepness. It tells you how much the y-coordinate of a point on the line changes for a one-unit change in the x-coordinate. To find the slope, you can use points on the line.
The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line.
Let's break down the steps: firstly, identify your points as \( (x_1, y_1) \) and \( (x_2, y_2) \). Next, plug their coordinates into the formula and solve.

• The numerator is \( y_2 - y_1 \).
• The denominator is \( x_2 - x_1 \).

Simplify both the numerator and denominator, and then divide to find the slope!

Understanding Coordinate Geometry

Coordinate geometry (or analytic geometry) allows us to describe geometric figures using algebraic formulas. It combines algebra and geometry to solve problems.
In this context, we're focusing on the coordinate system, which is a plane defined by an x-axis (horizontal) and a y-axis (vertical).

• Any point in this plane can be described using a pair of numbers \( (x, y) \).

These numbers are the coordinates of the point. The x-coordinate shows the horizontal position, while the y-coordinate shows the vertical position.
By using points on a line and the slope formula, we can understand the line’s behavior and characteristics. This helps in many practical applications like navigation, computer graphics, and physics.

Role of Algebraic Expressions

Algebraic expressions are at the heart of finding the slope. They allow us to create formulas that describe relationships between numbers.
In our slope calculation, we used the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \], which is itself an algebraic expression.
Let's dissect it further:

• \((y_2 - y_1)\) is the change in the y-coordinates, referred to as the rise.
• \((x_2 - x_1)\) is the change in the x-coordinates, known as the run.

Simplifying these expressions helps us determine the slope. Understanding basic algebra helps in solving not only slope but various other mathematical problems.
This is because algebra teaches us how to handle variables, constants, and operators systematically.

Steps in Problem Solving

Problem-solving in mathematics often involves sequential steps. Following a clear process helps in avoiding mistakes and ensures a logical workflow.
Here's a breakdown of the steps used to find the slope:

• Identify the Given Points: Recognize and note down the coordinates of the points given.
• Recall the Slope Formula: Bring to mind the slope formula to apply in the problem.
• Substitute the Values: Place the coordinates into the slope formula.
• Simplify the Expression: Ensure the formula’s terms are simplified for easier calculation.
• Compute the Slope: Perform the final arithmetic operation.

Practicing these problem-solving steps can improve mathematical skills. Always remember to double-check your work to avoid errors.