Problem 8 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-3,4),(2,-6)$$

Step-by-Step Solution

Answer

The slope of the line is -2.

Step 1: Understand the Formula for Slope

The slope \( m \) of a line is calculated using the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line.

Step 2: Identify the Coordinates

From the points \((-3, 4)\) and \((2, -6)\), identify \(x_1 = -3\), \(y_1 = 4\), \(x_2 = 2\), and \(y_2 = -6\).

Step 3: Substitute Values into the Slope Formula

Substitute the values into the slope formula: \( m = \frac{-6 - 4}{2 - (-3)} \). This simplifies to \( m = \frac{-6 - 4}{2 + 3} \).

Step 4: Calculate the Difference in y-coordinates

Calculate \(y_2 - y_1\): \(-6 - 4 = -10\).

Step 5: Calculate the Difference in x-coordinates

Calculate \(x_2 - x_1\): \(2 + 3 = 5\).

Step 6: Compute the Slope

Using the values from Step 4 and Step 5, compute the slope: \( m = \frac{-10}{5} \). This simplifies to \( m = -2 \).

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that involves plotting points on a plane to describe and solve geometric problems. It connects algebra and geometry, allowing us to analyze geometric shapes in an algebraic manner. This field uses a Cartesian coordinate system to specify the precise location of any point using ordered pairs

• The first number of the pair, the x-coordinate, indicates the position along the horizontal axis.
• The second number, the y-coordinate, indicates the position along the vertical axis.

For example, the point

• (-3, 4) means 3 units to the left of the origin and 4 units up.
• (2, -6) means 2 units to the right of the origin and 6 units down.

Understanding how points are placed on the coordinate plane helps us visualize and calculate various properties of lines and shapes, such as their slope, distance, or midpoints.

Slope Formula

The slope formula is a critical part of understanding the relationship between points in coordinate geometry. It provides a simple algebraic way to determine the steepness and direction of a line formed by any two points. The general formula for slope (\(m\) is:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

• This formula computes the change in the vertical direction (rise) over the change in the horizontal direction (run).
• A positive slope indicates an upward incline, while a negative slope suggests a downward decline.
• A zero slope means a horizontal line, and an undefined slope implies a vertical line.

Inserting the coordinates

• \((x_1, y_1) = (-3, 4)\) and \((x_2, y_2) = (2, -6)\)

into the slope formula helps find the line's slope, offering a clear depiction of how steep the line is and in which direction it moves across the plane.

Algebraic Calculation

Algebraic calculation involves using rules and techniques of algebra to solve mathematical problems. For calculating the slope, the algebraic process includes substituting known values into the slope formula and simplifying. Let's break it down:

• First, identify the given points: \((-3, 4)\) and \((2, -6)\).
• Substitute these into the formula: \(m = \frac{-6 - 4}{2 - (-3)}\).

Next, simplify the expression by performing the arithmetic operations:

• Calculate \(y_2 - y_1\): which results in \(-10\).
• Calculate \(x_2 - x_1\): yielding \(5\).

Then complete the division:

• \(m = \frac{-10}{5}\), which simplifies to \(-2\).

This algebraic calculation tells us the slope of the line is \(-2\), indicating a downward slope as you move from left to right across the coordinate plane.