Problem 12 - Elementary Algebra

Question

Find the slope of the line determined by each pair of points. $$(-3,-6),(5,-6)$$

Step-by-Step Solution

Answer

The slope is 0.

Step 1: Review the Slope Formula

The slope of a straight line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( m \) represents the slope.

Step 2: Plug in the Coordinates

Substitute the coordinates of the given points \((-3, -6)\) and \((5, -6)\) into the slope formula: \[ m = \frac{-6 - (-6)}{5 - (-3)} \]

Step 3: Simplify the Numerator

Calculate the difference in the \( y \)-coordinates: \[ -6 - (-6) = -6 + 6 = 0 \] So the formula simplifies to: \[ m = \frac{0}{5 - (-3)} \]

Step 4: Simplify the Denominator

Calculate the difference in the \( x \)-coordinates: \[ 5 - (-3) = 5 + 3 = 8 \] So the formula becomes: \[ m = \frac{0}{8} \]

Step 5: Calculate the Slope

Any number divided by 8 results in 0: \[ m = 0 \] Therefore, the slope of the line is 0.

Key Concepts

Slope Formula

The slope formula is an essential tool in coordinate geometry to measure how steep a line is. If you're given two points on a line, such as \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope \( m \) can be calculated using the following equation:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
The numerator \( (y_2 - y_1) \) represents the change in vertical position, or how much the line goes up or down between the two points. The denominator \( (x_2 - x_1) \) describes the change in horizontal position, or how far left or right the line moves.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• A zero slope indicates a horizontal line, meaning there is no vertical change between the points.
• An undefined slope occurs when the denominator is zero, indicating a vertical line with no horizontal change.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometry using a coordinate system. In coordinate geometry, points are placed on a graph using ordered pairs of numbers that indicate their position.
Each point is represented as \( (x, y) \), where \( x \) is the horizontal coordinate (how far the point is along the horizontal axis, or x-axis) and \( y \) is the vertical coordinate (how far up or down the point is along the vertical axis, or y-axis).
Using this system, you can represent geometric figures like lines and curves, which can then be analyzed mathematically. A line in coordinate geometry is defined by a linear equation, which can be expressed in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

• Coordinate geometry provides a bridge between algebraic and geometric concepts.
• It allows precise calculations of distances, midpoints, and slopes, making it easier to solve complex problems.
• It's widely used in various fields including physics, engineering, and computer graphics.

Calculating Slopes

Calculating the slope of a line between two points is a straightforward process using the slope formula. Let's break it down further:
1. **Identify the Coordinates:** Start by identifying the two points the line passes through. For example, \( (-3, -6) \) and \( (5, -6) \).
2. **Plug into the Formula:** Insert these values into the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( y_1 = -6, y_2 = -6, x_1 = -3, x_2 = 5 \).
3. **Calculate the Differences:** Compute the difference in y-coordinates: \( y_2 - y_1 = -6 - (-6) = 0 \). Then compute the difference in x-coordinates: \( x_2 - x_1 = 5 - (-3) = 8 \).
4. **Find the Slope:** Simplify the equation: \( m = \frac{0}{8} = 0 \).

• The slope between these two points is 0, indicating a horizontal line.
• Understanding the steps involved in slope calculation helps intuitively grasp line orientation and behavior.
• Practicing with different points helps solidify the concept and improve problem-solving skills.