Problem 58 - Elementary Algebra

Question

Find the coordinates of two points on the given line, and then use those coordinates to find the slope of the line. $$y=3$$

Step-by-Step Solution

Answer

The slope of the line \(y = 3\) is 0.

Step 1: Identify the Type of Line

The equation given, \(y = 3\), describes a horizontal line. This means that all points on the line have the same \(y\)-coordinate, which is 3.

Step 2: Choose Points on the Line

Select any two points that satisfy the equation \(y = 3\). For simplicity, choose \((0, 3)\) and \((2, 3)\). These are arbitrary points where the \(x\)-coordinate can be any real number, but the \(y\)-coordinate must always be 3.

Step 3: Calculate the Slope of the Line

The formula for the slope, \(m\), is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((0, 3)\) and \((2, 3)\):\[m = \frac{3 - 3}{2 - 0} = \frac{0}{2} = 0\]The slope of this horizontal line is 0.

Key Concepts

Horizontal Line

A horizontal line is one of the simplest forms of line equations you will encounter. In any equation of the form \(y = c\), where \(c\) is a constant, the line is horizontal.
The key characteristic of a horizontal line is that every point on the line shares the same y-coordinate. For instance, in the equation \(y = 3\), all points have a y-coordinate of 3, meaning the line runs parallel to the x-axis.
Because of its parallel nature to the x-axis, the line extends infinitely left and right. This results in a line that does not rise or fall regardless of the x-value you choose.
In terms of functionality, such lines reflect scenarios where a value remains unchanged regardless of another influencing variable. This can model real-life situations like a flat terrain height or a constant cost.

Slope Calculation

Calculating the slope of a line is an important concept in geometry that describes the line's steepness. For any two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula to find the slope \(m\) is:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
In our example with the horizontal line \(y = 3\), pick any two points such as \((0, 3)\) and \((2, 3)\). Substituting these points into the slope formula yields:

• Difference in \(y\)-coordinates: \(3 - 3 = 0\)
• Difference in \(x\)-coordinates: \(2 - 0 = 2\)

The calculation becomes \(\frac{0}{2} = 0\). This results in a slope of 0, confirming the notion that horizontal lines are perfectly flat.
A zero slope signifies no vertical change, meaning the line doesn't rise or fall as it moves horizontally.
Learning to see this visually on a graph can help reinforce the concept further.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves understanding geometric figures through algebraic equations. It serves as a bridge connecting algebra to geometry using a coordinate plane.
When dealing with equations like \(y = c\), recognizing the type of line they represent is important. You select random x-values on a coordinate plane and evaluate the consistent y-value dictated by the equation.
For horizontal lines, once the equation is recognized, you can plot any number of points, each showing the same y-value in response to a variety of x-values.

• Key features required include being comfortable with plotting points using an x, y-axis framework.
• Familiarity with terms like slope, which describes line inclination, further enhances comprehension.

Overall, mastering coordinate geometry allows us to map real-life dynamics simply by understanding the interactions of algebraic expressions. It forms a base for solving complex problems by transforming algebraic representations into visualizations through plotting on a coordinate grid.