Problem 15 - Algebra 2

Question

Find the slope of the line through each pair of points. \((-2,-1)\) and \((8,-3)\)

Step-by-Step Solution

Answer

The slope of the line through the points (-2, -1) and (8, -3) is \( -\frac{1}{5} \)

Step 1: Identify the Points

First, identify the given pairs of points. Here, we have Point 1 with coordinates (-2, -1) and Point 2 with coordinates (8, -3).

Step 2: Use the Slope Formula

To find the slope (m) of the line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Plug in the Coordinates

Substitute the coordinates of the two points into the slope formula: \[ m = \frac{-3 - (-1)}{8 - (-2)} = \frac{-3 + 1}{8 + 2} = \frac{-2}{10} \. \]

Step 4: Simplify the Slope

Simplify the fraction to obtain the slope of the line: \[ m = \frac{-2}{10} = -\frac{1}{5} \. \]

Key Concepts

Slope Formula

Understanding the slope of a line is crucial for navigating the basics of coordinate geometry and linear equations. The slope, denoted as 'm', measures the steepness or incline of a line. In numerical terms, it's the change in the vertical direction (rise) over the change in the horizontal direction (run) between any two points on a line.

To calculate the slope between two points, you can use the following slope formula:
\[\begin{equation}m = \frac{y_2 - y_1}{x_2 - x_1}\end{equation}\]
To apply this, simply subtract the y-coordinate of the first point from the y-coordinate of the second point (\(y_2 - y_1\)), and divide by the difference between their x-coordinates (\(x_2 - x_1\)). For the exercise at hand, with points \((-2,-1)\) and \((8,-3)\), plugging the values into the slope formula, we get \(-\frac{2}{10}\), which simplifies to \(-\frac{1}{5}\), indicating a slight downward tilt of the line on the graph.

Coordinate Geometry

In coordinate geometry, which is also known as analytic geometry, we study geometrical shapes and figures using the coordinate plane. The basic principle involves plotting points, lines, and curves onto the plane, and then using algebraic formulas to analyze their properties like length, area, and slope.

Within this framework, the concept of computing a line's slope links together algebra and geometry, offering a way to quantify the direction and steepness of the line. When you know the coordinates of two points on a line, as in our exercise with points \((-2,-1)\) and \((8,-3)\), you have sufficient information to calculate the line's slope. This information is pivotal when it comes to understanding how lines relate to each other; for instance, parallel lines have equal slopes, while perpendicular lines have slopes that are negative reciprocals.

Linear Equations

Linear equations form the backbone of algebra and can be graphically represented as straight lines on the coordinate plane. These equations generally take the form \(y = mx + b\), where 'm' is the slope and 'b' is the y-intercept, the point where the line crosses the y-axis. Knowing how to find a line's slope is an integral part of solving and graphing these equations.

In our example, if we were to write an equation for the line passing through \((-2,-1)\) and \((8,-3)\), the slope \(-\frac{1}{5}\) would be 'm' in the equation. This tells us that for every unit we go right (the positive direction along the x-axis), the line goes down \(\frac{1}{5}\) of a unit. Understanding this relationship between the algebraic equation and the geometric representation on the graph helps students visualize and solve problems involving linear equations.