Problem 61 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (-1,-7),(-2,-9) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 2.

Step 1: Identify the coordinates of the points

The coordinates of the points are given as: Point 1: (-1, -7) Point 2: (-2, -9)

Step 2: Plug in the coordinates into the slope formula

Using the slope formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Substitute the coordinates from Step 1: $$ m = \frac{-9 - (-7)}{-2 - (-1)} $$

Step 3: Simplify the expression

Now we simplify the expression: $$ m = \frac{-9 + 7}{-2 + 1} $$ $$ m = \frac{-2}{-1} $$

Step 4: Calculate the slope

Finally, we divide the numbers to get the slope: $$ m = 2 $$ The slope of the line through the pair of points (-1, -7) and (-2, -9) is 2.

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is where algebra meets geometry. It involves placing geometric figures in a coordinate plane and using numerical methods to solve geometric problems. Imagine a graph with a horizontal x-axis and a vertical y-axis. This grid, called the Cartesian plane, allows us to locate points using pairs of numbers known as coordinates.
Each point on this plane has an x-coordinate and a y-coordinate, like in our problem where we had Point 1 as each point on this plane has an x-coordinate and a y-coordinate, like in our problem where we had Point 1 as (-1, -7) and Point 2 as (-2, -9). These coordinates define specific points on the plane. Using these coordinates, you can analyze the relationships, such as distance or slope, between points.
Coordinate geometry gives you the tools to translate geometric problems into algebraic equations. This not only allows for easier calculations but also provides a way to visualize algebraic equations through graphs and shapes. Whether you're finding the slope of a line or the distance between points, knowing coordinate geometry is essential.

Demystifying Linear Equations

Linear equations are mathematical expressions that represent straight lines when plotted on a graph. The general form of a linear equation in two dimensions is:\[ ax + by = c \] Where \(a\), \(b\), and \(c\) are constants. In simpler terms, linear equations create straight lines, making them predictable and easy to understand.
The slope, which is what we found in the exercise, is a crucial part of describing a linear equation. It's the "rise over run," indicating how steep a line is on the graph. In the slope-intercept form, \(y = mx + c\), \(m\) represents the slope, and \(c\) is the y-intercept, the point where the line crosses the y-axis.
Understanding linear equations can help you make predictions. For instance, if you know the slope of a line, you can forecast how a change in \(x\) affects \(y\), which is especially useful in real-world applications like economics or physics.

The Significance of Pairs of Points

Pairs of points play a fundamental role in coordinate geometry. They allow us to determine the unique properties of lines, such as slope and distance. Each pair consists of two distinct points, which when plotted, can help define a line.
In the given problem, the pair of points (-1, -7) and (-2, -9) were used to calculate the slope. The formula for slope is crucial here: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula essentially measures the change in \(y\) (vertical) over the change in \(x\) (horizontal) between two points.
Using pairs of points not only helps in finding the slope but can also determine if two lines are parallel or perpendicular, aiding in understanding the geometric structure of graphs. By mastering the concept of pairs of points, you gain an essential skill for solving many problems in both algebra and geometry.