Problem 47 - Elementary Algebra

Question

Find the slope, if it exists, of the line through the given pairs of points. $$ (8,-2), \quad(10,-6) $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -2.

Step 1: Identify the coordinates of the given points

We are given two points: (8, -2) and (10, -6). We will label the coordinates as follows:$$ (x_1, y_1) = (8, -2) \\ (x_2, y_2) = (10, -6). $$

Step 2: Plug the coordinates into the slope formula

Now that we have the coordinates labeled, we can plug them into the slope formula:$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$Substitute the coordinates:$$ m = \frac{-6 - (-2)}{10 - 8} $$

Step 3: Simplify the equation

Now, we will simplify the equation to find the slope m:$$ m = \frac{-6 + 2}{2} \\ m = \frac{-4}{2} \\ m = -2. $$

Step 4: State the slope of the line

The slope of the line that passes through the given points (8, -2) and (10, -6) is$$ m = -2. $$

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that involves the study of geometric figures through coordinates and algebraic equations. It combines elements of algebra and geometry, allowing us to precisely describe the positions of points and the shapes of figures using mathematical equations.
In the plane, we use a coordinate system with two axes, usually labeled as the x-axis and y-axis.

• The x-axis is horizontal and increases from left to right.
• The y-axis is vertical and increases from bottom to top.

Every point in the plane is represented by an ordered pair, \(x, y\), where the x-coordinate shows the horizontal position and the y-coordinate shows the vertical position. This system is invaluable in analyzing shapes, distances, and slopes within the plane.

Slope Formula

The slope of a line in coordinate geometry is a measure of its steepness and direction. It is defined as the ratio of the change in the y-coordinates to the change in the x-coordinates between two points on a line. This is mathematically represented by the slope formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here:

• \( m \) denotes the slope of the line.
• \( x_1, y_1 \) and \( x_2, y_2 \) are the coordinates of two distinct points on the line.

The slope tells us how much the y-coordinate changes for a unit change in the x-coordinate. In simpler terms, if you imagine walking along the line, the slope indicates whether you're going uphill, downhill, or staying level. If the slope is positive, the line ascends; if the slope is negative, it descends. When the slope is zero, the line is horizontal, and when undefined, it's vertical.

Linear Equations

Linear equations represent straight lines on a coordinate plane. These equations are typically in the form:\[y = mx + c\]where:

• \( y \) and \( x \) are variables.
• \( m \) is the slope of the line.
• \( c \) is the y-intercept, which is the point where the line crosses the y-axis.

Linear equations can be derived when we know the slope and a point on the line, or the y-intercept and the slope. They are fundamental to understanding relationships and trends within data, as they can model directly proportional relationships. Solving these equations involves finding x-values that satisfy the equation for given y-values or vice versa, often requiring skills in algebra and understanding of coordinate geometry.