Problem 21 - Elementary Algebra

Question

Graph the equations. $$ y-2=0 $$

Step-by-Step Solution

Answer

Answer: The equation represents a horizontal line, and its y-intercept is at point (0,2).

Step 1: Identify the equation format

The given equation is "y-2=0". We can rewrite this equation as "y=2". This represents a horizontal line where the y-coordinate is always 2.

Step 2: Determine the y-intercept

A horizontal line will always have the same y-coordinate for any x-coordinate. In this case, the y-coordinate is 2 which means the y-intercept will be at the point (0,2).

Step 3: Plot the y-intercept

On the coordinate plane, locate the point where x=0 and y=2. This is the y-intercept of the line, so plot this point on the graph.

Step 4: Draw the horizontal line

Since we have identified that this is a horizontal line and have plotted the y-intercept, we can draw the line by simply extending it in both directions (left and right) from the y-intercept. Ensure that it remains parallel to the x-axis and that the y-coordinate stays at 2 for all points on the line. The graph of the equation $$y-2=0$$ or $$y=2$$ is a horizontal line with y-intercept at (0,2).

Key Concepts

Coordinate Plane

The coordinate plane, often known as the Cartesian plane, is a two-dimensional surface where each point is determined by a pair of numerical coordinates. These coordinates are, essentially, the 'address' of the point on the flat plane. The plane is divided into four quadrants by two perpendicular lines: the horizontal x-axis and the vertical y-axis.

The point where these axes intersect is called the origin, denoted as (0, 0). To locate a point on this plane, you simply move along the x-axis to reach the appropriate x-coordinate, and then vertically along the y-axis to reach the y-coordinate. Every point on the plane can be described using this system, allowing for precise plotting of lines, shapes, and functions.

Y-Intercept

In the context of graphing lines, the y-intercept is a fundamental concept. It refers to the point where the line crosses the y-axis. To be precise, it's the coordinate at which the x-value is zero. For every linear equation in the form of 'y = mx + b', the 'b' value represents the y-intercept, showing the point at which the line will intersect the y-axis.

The y-intercept is crucial for graphing because it provides a starting point from which the line can be extended in a straight path adhering to its slope. In a horizontal line equation, like in our example 'y = 2', the y-intercept doesn't vary with x, which means the y-coordinate stays constant and the line will be horizontal at that y-coordinate across the graph.

Horizontal Line Equation

A horizontal line equation takes the form 'y = c' where 'c' is a constant. This tells us that no matter the value of x, the value of y will always be the same. In the exercise 'y - 2 = 0', simplifying to 'y = 2' gives us a horizontal line. This equation implies that every point on the line has a y-coordinate of 2, regardless of the x-coordinate.

When plotting this on a coordinate plane, you only need one point to establish the horizontal line because of its constant y-coordinate. After marking your initial point, the rest of the line is drawn to the left and right, maintaining the same distance from the x-axis. This results in a straight line, parallel to the x-axis that extends infinitely in both directions.