Problem 12 - Algebra 2

Question

Find the slope of the line through each pair of points. \((-3,9)\) and \((0,3)\)

Step-by-Step Solution

Answer

The slope of the line through the points (-3, 9) and (0, 3) is -2.

Step 1: Identify Coordinates

Identify the coordinates of the two points. Point A is (-3, 9), which means that for A, the x-coordinate is -3 and the y-coordinate is 9. Point B is (0, 3), with B's x-coordinate as 0 and y-coordinate as 3.

Step 2: Calculate the Change in Y

Calculate the change in y-coordinates, also known as the rise, by subtracting the y-coordinate of point A from the y-coordinate of point B. The change in y-coordinates (\text{\(\Delta y\)}) is calculated as \text{\(3 - 9 = -6\)}.

Step 3: Calculate the Change in X

Calculate the change in x-coordinates, also known as the run, by subtracting the x-coordinate of point A from the x-coordinate of point B. The change in x-coordinates (\text{\(\Delta x\)}) is calculated as \text{\(0 - (-3) = 3\)}.

Step 4: Compute the Slope

Divide the change in y-coordinates by the change in x-coordinates to find the slope. Using the formula for slope, \text{\(\frac{\Delta y}{\Delta x}\)}, plug in the values to get \text{\(\frac{-6}{3} = -2\)}. Thus, the slope of the line is -2.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach combines algebra and geometry to describe the position, shape, and size of geometric figures on a plane.

When working with coordinate geometry, points are located on a grid, known as the Cartesian coordinate system, using an ordered pair of numbers \( (x, y) \). These pairs represent the horizontal (x-axis) and vertical (y-axis) distances from the origin, \( (0, 0) \). Understanding how to plot these points and interpret their positions is fundamental when we're finding the slope of a line, as it involves seeing how far along and up or down the points are from each other on the coordinate plane.

Slope Formula

The slope of a line in coordinate geometry represents the steepness and direction of the line. To calculate it, we use the slope formula, which is \( m = \frac{\Delta y}{\Delta x} \), where \( m \) is the slope, \( \Delta y \) is the change in y-coordinates (rise), and \( \Delta x \) is the change in x-coordinates (run).

Direct Comparison

By comparing two points on a line, such as Point A \( (-3, 9) \) and Point B \( (0, 3) \), and using their coordinates in the formula, we can determine how much the line rises or falls over a given horizontal distance. This ratio is essential for understanding the behavior of the line on a graph and is a critical component of many algebraic concepts.

Rate of Change

In mathematics, the concept of rate of change is a way of describing how a certain quantity changes with respect to another. Slope is a specific type of rate of change; it measures the change in y (dependent variable) with respect to the change in x (independent variable).

Understanding Through Slope

A slope with a greater absolute value indicates a steeper line, meaning for each unit of x, the value of y increases or decreases more quickly. Conversely, a slope closer to zero indicates a flatter line. This concept is critical when analyzing how fast something is moving or changing over time, making it a powerful tool in various scientific and engineering fields.

Linear Equations

Linear equations form the basis for lines in coordinate geometry. They can be written in various forms, including slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Slope as a Key Component

The slope within a linear equation indicates how the line tilts relative to the horizontal axis, and it plays a significant role in the graphical representation of the equation. By finding the slope, as in our example with points \( (-3, 9) \) and \( (0, 3) \), we grasp how the variables in the equation relate to one another and change concurrently, which is crucial for solving and graphing linear relationships.