Problem 26 - Elementary Algebra

Question

For the following problems, graph the equations. $$ y=3 $$

Step-by-Step Solution

Answer

Answer: The two points identified on the graph of the equation y = 3 are (0, 3) and (1, 3).

Step 1: Choose Values for x

Choose any two values for x. For example, we can choose x = 0 and x = 1.

Step 2: Find the Corresponding y Values

Since the equation is given as y = 3, the y-coordinate will always be equal to 3. So, for x = 0 and x = 1, we have y = 3.

Step 3: Identify the Points on the Graph

From the previous steps, we have the points (0, 3) and (1, 3).

Step 4: Plot the Points on the Graph

Place the points (0, 3) and (1, 3) on the graph.

Step 5: Draw the Line

Connect the points (0, 3) and (1, 3) with a straight line. This line represents the equation y = 3.

Key Concepts

Plotting Points on a Graph

When plotting the equation y = 3, we navigate through a fundamental skill of graphing which is plotting points. Imagine the coordinate system as a map, each point defined by a pair of numbers telling us where to find it.

To plot a point, you begin at the origin, where the x and y axes intersect. From there, move right if the x-value is positive, left if it's negative, up for a positive y-value, and down if it's negative. Once you reach the correct location along the x-axis, move parallel to the y-axis to locate the exact point. For the equation y = 3, regardless of the x-value you choose, the y-value remains consistently at 3.

Choosing Points

In the step-by-step solution, x-values of 0 and 1 were chosen arbitrarily because they are simple to work with and demonstrate the concept effectively. The resulting points, (0, 3) and (1, 3), are both located three units above the x-axis, firmly planting them on the horizontal line where y = 3.

Horizontal Lines

If we look at the equation y = 3 closely, we see it's a special type of linear equation that creates a horizontal line on the graph. In a horizontal line, every point has the same y-coordinate, which shows that there is no vertical change as you move along the line.

In this case, all points on the line have a y-coordinate of 3, hence, the line is drawn parallel to the x-axis, and three units above it. This is true for any value you pick for x. The constancy of y in the equation reflects the 'flatness' of the line - it does not slope upwards or downwards.

Identifying Horizontal Lines

To identify a horizonal line, remember that its equation will always be in the form of y = k, where k is any constant number. Here, k is 3. The graph of this equation will be a straight line that cuts across the y-axis at the point y = k.

Coordinate System

The coordinate system, commonly known as the Cartesian coordinate system is a two-dimensional plane consisting of a horizontal axis (x-axis) and a vertical axis (y-axis). These axes intersect at a point called the origin, which has the coordinates (0, 0).

The coordinate system allows us to precisely locate points and draw various shapes, including lines, using pairs of numbers called coordinates. Each point is represented by a pair (x, y), where x is the position relative to the vertical y-axis and y is the position relative to the horizontal x-axis.

Understanding the Quadrants

The system is divided into four quadrants, with each quadrant corresponding to a unique combination of positive and negative values for x and y. This standardized system is essential for graphing equations, such as y = 3, and understanding their graphical implications. Without this system, expressing and visualizing mathematical concepts would be exceedingly difficult.