Problem 18 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y=3 $$

Step-by-Step Solution

Answer

Graph a horizontal line at \( y = 3 \).

Step 1: Understand the Equation

The equation given is \( y = 3 \). This is an equation of a horizontal line, where the value of \( y \) is always 3.

Step 2: Identify Points on the Line

For a horizontal line \( y = 3 \), choose several \( x \) values to plot. It doesn't matter what \( x \) values you select, as \( y \) will always equal 3. For example, you could choose \( x = -2, 0, 2 \).

Step 3: Plot the Points

Plot the points based on your selection: \((-2, 3)\), \((0, 3)\), and \((2, 3)\). These all lie on the horizontal line where \( y = 3 \).

Step 4: Draw the Line

Connect the points with a straight line that extends to the edges of your graphing area. This line represents all solutions to the equation \( y = 3 \). It is parallel to the x-axis.

Key Concepts

Horizontal Line

Understanding a horizontal line is essential when graphing linear equations. A horizontal line on a graph means that all points on this line have the same y-value. In the equation y = 3, the line is specifically placed where y is always 3, regardless of the x-value.

• Horizontal lines are parallel to the x-axis.
• These lines have a slope of zero, indicating there is no vertical change as we move along the line.
• In equations of the form y = k, k represents the consistent y-value for all points on the line.

To visualize it, imagine a line slicing through the coordinate plane horizontally at y = 3. This characteristic differentiates it from vertical lines or diagonal ones, which have varying slopes.

Coordinate Plane

The coordinate plane is like a map where you can plot points and graph equations. It consists of a horizontal line called the x-axis and a vertical line called the y-axis. These axes intersect at the origin, labeled as (0, 0).

• Each point on the plane is defined by a pair of numbers (x, y).
• The x-value indicates horizontal position, while the y-value shows vertical position.
• The coordinate plane is divided into four quadrants by the axes.

When dealing with linear equations, such as y = 3, the entire line fits within this plane and stretches horizontally across it. Understanding the layout of the coordinate plane makes pinpointing and plotting each point easier, allowing for graphical solutions to equations like y = 3.

Graphing Techniques

Graphing linear equations can be straightforward if you follow a systematic approach. Here’s how to tackle plotting an equation like y = 3:

• Start by recognizing the type of line based on the equation format.
• Select any values for x; the y-value remains unchanged in y = 3.
• Mark points on the graph using chosen x-values and the constant y-value.
• Draw a straight line through these points, extending across the graph.

This approach simplifies graphing by focusing on consistency. In horizontal line scenarios, like y = 3, the key technique is recognizing the unchanging y-value across all plotted points. This ensures the plotted line accurately represents the solution set defined by the equation.