Problem 62 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (-6,-6),(-5,-4) $$

Step-by-Step Solution

Answer

Answer: The slope of the line passing through the points (-6, -6) and (-5, -4) is 2.

Step 1: Identify the Given Points

The given points are \((-6, -6)\) and \((-5, -4)\). Let these points be \((x_1, y_1)\) and \((x_2, y_2)\), respectively.

Step 2: Apply the Slope Formula

We will use the slope formula to find the slope of the line: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

Step 3: Substitute the Values

Now, let's substitute the values of the points into the formula: $$ m = \frac{-4 - (-6)}{-5 - (-6)} $$

Step 4: Simplify the Equation

Simplify the numerator and the denominator: $$ m = \frac{2}{1} $$

Step 5: Calculate the Slope

As we have simplified the equation, the slope of the line is: $$ m = 2 $$ Therefore, the slope of the line passing through the points \((-6, -6)\) and \((-5, -4)\) is 2.

Key Concepts

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is the study of geometry using a coordinate system. In this system, points are defined by their location along the x (horizontal) and y (vertical) axes.
With coordinate geometry, each point is represented as a pair \(x, y\). For instance, the point \((-6, -6)\) is located 6 units left and 6 units down from the origin, which is \((0, 0)\).
Coordinate geometry is fundamental because it allows us to use algebra to solve geometric problems. It forms a bridge between algebra and geometry, and this connection helps in understanding the properties and relationships of geometric figures.

Slope Formula

The slope is a measure of how steep a line is. It is calculated using the slope formula, a key part of coordinate geometry. The formula is written as:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This equation gives the slope \(m\) of a line that passes through two points \(x_1, y_1\) and \(x_2, y_2\). Here, \(y_2 - y_1\) represents the vertical change between the points, known as the 'rise', while \(x_2 - x_1\) represents the horizontal change, known as the 'run'.
A positive slope means the line rises as it moves from left to right. A negative slope means it falls. A zero slope means the line is horizontal, and an undefined slope indicates a vertical line. Understanding how to apply and interpret the slope formula is crucial in coordinate geometry.

Mathematics Calculations

Let's break down the mathematics calculations needed for finding the slope using the given exercise. We have two points: \((-6, -6)\) and \((-5, -4)\). These are our \(x_1, y_1\) and \(x_2, y_2\).
To find the slope:

• Substitute the values into the slope formula: \(m = \frac{-4 - (-6)}{-5 - (-6)}\)
• Solve the operations: \(m = \frac{2}{1}\)

By simplifying the fraction \(\frac{2}{1}\), we have our slope, which is 2. This tells us the line increases by 2 units vertically for every 1 unit it moves horizontally.
Understanding how to substitute and simplify these expressions is important in accurately calculating the slope in coordinate geometry problems.