Problem 31 - Prealgebra

Question

Find three solutions to each of the equations and use them to draw the graph. (GRAPH CANT COPY) $$x=4$$

Step-by-Step Solution

Answer

The equation \(x=4\) is a vertical line through \(x=4\). Possible solutions include \((4, 0), (4, 1), (4, 2)\).

Step 1: Understand the Equation

The given equation is \(x = 4\), which represents a vertical line in the coordinate plane. This line consists of all points where the \(x\)-coordinate is 4, regardless of the \(y\)-coordinate.

Step 2: Identify Three Solutions

To find three solutions, we need to choose three different \(y\) values while keeping \(x = 4\). For example, we can select \(y = 0, 1, 2\). Then, the solutions are: \((4, 0), (4, 1), (4, 2)\).

Step 3: Draw the Graph

Using the three solutions \((4, 0), (4, 1), (4, 2)\), draw a vertical line on the graph where each point lies on the line where \(x = 4\). This line extends indefinitely in the vertical direction through all integer and non-integer \(y\) values.

Key Concepts

Vertical Line

A vertical line is a straight line that runs from top to bottom on the graph. In mathematics, when we say a line is vertical, we mean that it does not slant to the right or left. It stays perfectly upright. For any vertical line, all points on the line have the same x-coordinate. This is an essential feature that distinguishes vertical lines from other lines, such as horizontal or diagonal lines.

In the problem, the equation given is \(x = 4\). This specifies a vertical line on the coordinate plane. Every point on this line shares the x-coordinate of 4, while the y-coordinates can be any real number. So, if you imagine this line, it goes straight up and down at the x-axis point of 4.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves. It is divided into four quadrants by the horizontal axis (x-axis) and the vertical axis (y-axis).

In the coordinate plane, each point is defined by a pair of numbers (x, y). These numbers tell you the point's location relative to the center, known as the origin. The origin has coordinates (0, 0). Understanding the layout of the coordinate plane is crucial for plotting equations and finding intersections.

When dealing with the equation \(x = 4\), we visualize its graph as a line on the coordinate plane. This helps in understanding how lines behave based on changes in the equation's variables.

Finding Solutions

Finding solutions in a linear equation involves figuring out which points satisfy the equation. For the equation \(x = 4\), the process is straightforward because the x-coordinate is fixed.

To discover three solutions, you choose any three different values for y, keeping x constant. For instance:

• When \(y = 0\), the point (4, 0) is a solution.
• If \(y = 1\), the point (4, 1) works.
• When \(y = 2\), the point (4, 2) is another solution.

This method confirms that the line passes through multiple points, proving it's a valid solution.

X-Coordinate

The x-coordinate is the first number in a coordinate pair (x, y), and it tells you how far along the x-axis a point is. In the equation \(x = 4\), the x-coordinate is always 4, no matter what the y-coordinate is.

This consistent x-coordinate is why the graph of \(x = 4\) forms a vertical line. By maintaining the same x-value, all the points sit directly above each other. Hence, once plotted on the coordinate plane, they align vertically. Knowing the importance of the x-coordinate aids in correctly graphing and understanding the behavior of equations.

Y-Coordinate

The y-coordinate is the second number in a coordinate pair (x, y), indicative of a point's position along the y-axis. Compared to the x-coordinate in the given equation \(x = 4\), the y-coordinate can vary vastly, leading to a spread of points vertically.

In our solutions (<4, 0>, <4, 1>, <4, 2>), you selected different y-values to show the points that lie on the line. Variations in the y-coordinate illustrate the expansion and contraction of the linear graph along the vertical dimension.

Thus, manipulating the y-coordinate while keeping the x-coordinate constant is key to exploring the full vertical extent of the graph line. The interplay between x and y in coordinate pairs is critical to understanding how they structure the graph.