Problem 63 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. $$ (-1,0),(-2,-2) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 2.

Step 1: Identify the given points

We have two points (-1, 0) and (-2, -2). Let (-1, 0) be \((x1, y1)\) and (-2, -2) be \((x2, y2)\).

Step 2: Write down the formula for finding the slope of a line

The formula for the slope of a line given two points is: $$ m = \frac{(y2 - y1)}{(x2 - x1)} $$

Step 3: Plug in the values of the coordinates into the formula

We have \((x1, y1) = (-1, 0)\) and \((x2, y2) = (-2, -2)\). Plugging these values into the formula to calculate the slope, we get: $$ m = \frac{(-2 - 0)}{(-2 - (-1))} $$

Step 4: Calculate the slope

Now we can simplify and calculate the slope: $$ m = \frac{-2}{-1} = 2 $$

Step 5: Final Answer

The slope of the line through the pair of points \((-1, 0)\) and \((-2, -2)\) is 2.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where numerical coordinates are used to specify the position of points on a plane. It's like a bridge that combines algebra and geometry, allowing us to describe geometric figures using equations. Let's consider a few fundamental aspects:

• Coordinates: These are pairs of numbers written in the form \(x, y\), representing a point's position on the Cartesian plane.
• Cartesian Plane: A two-dimensional plane divided by a horizontal line, the x-axis, and a vertical line, the y-axis.
• Distance and Midpoints: Coordinate geometry helps to calculate distances between points and find midpoints by utilizing specific formulas.

We often use coordinate geometry to mathematically represent real-world scenarios, putting points on the plane and using them to understand slopes and lines. For instance, finding the slope of a line through two points requires such a representation.

Linear Equations

Linear equations form one of the basic concepts linking algebra and geometry. They represent straight lines on a graph, described by the equation \(y = mx + b\), where:

• \(m\) is the slope, which indicates the steepness or incline of the line. It is calculated as the change in \(y\) over the change in \(x\).
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

The slope of a line is central to understanding linear equations because it tells us how the line changes as we move along the x-axis. For example, in our problem, the slope of the line connecting points \((-1,0)\) and \((-2,-2)\) is \(2\), showing us how the y-coordinates change in relation to the x-coordinates as we move between these points. Recognizing the role of the slope helps us predict how one variable changes as the other changes.

Mathematical Problem Solving

Mathematical problem solving involves a series of logical steps to find an answer to a given problem, especially in coordinate geometry and linear equations. Here’s how we did it for finding the slope:

Identify the Points

We gathered relevant data by identifying point coordinates \((-1,0)\) and \((-2,-2)\). This step is crucial as it provides the necessary inputs for calculations.

Use the Formula

Once the data is ready, apply the formula for slope: \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). Input the points into the formula.

Calculate the Result

Substitute values and simplify the expression to find the slope, \(m = \frac{-2}{-1} = 2\). Therefore, the line’s slope is 2.Effective problem solving means breaking down the problem into manageable steps, helping to avoid confusion and errors. Practices like these enable students to approach coordinate geometry and algebraic problems methodically.