Problem 57 - Precalculus Essentials

Question

Graph equation in a rectangular coordinate system. $$3 x-18=0$$

Step-by-Step Solution

Answer

The graph of the equation \(3x - 18 = 0\) is a vertical line that passes through the x-axis at \(x = 6\).

Step 1: Rearrange the Equation

The given equation is \(3x - 18 = 0\). We can rewrite this equation to get \(x\) alone on one side. This is done by adding 18 to both sides of the equation and then dividing both sides by 3 and we get \(x = 6\).

Step 2: Plot the Graph

The equation \(x = 6\) represents a vertical line passing through the point (6, 0) on the x-axis, as well as through every other point where \(x = 6\). In other words, the line passes through points such as (6, 1), (6, -1), (6, 2), etc. Therefore, plot a vertical line that passes through these points.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system is a fundamental concept used in graphing equations. It provides a grid to help visualize mathematical equations and data. This system is composed of two perpendicular number lines that intersect at their zero points, known as the origin.
The horizontal line is known as the x-axis, and the vertical line is known as the y-axis. These axes split the plane into four quadrants, which help in locating any point with ordered pairs \((x, y)\).

• The first quarter is where both coordinates are positive.
• The second quarter features a negative x and a positive y.
• The third quarter has both coordinates negative.
• The fourth has a positive x and a negative y.

Understanding these divisions is crucial when plotting points or graphing lines, as it helps relate an algebraic equation to a visual graph.

Vertical Lines

In the rectangular coordinate system, vertical lines are a special type of line where all points share the same x-coordinate. This means that a vertical line does not rely on the y-coordinate to position its points, as all changes occur perpendicular to the x-axis.
For example, a graph of the equation \(x = 6\) results in a vertical line crossing through the x-axis at 6. This line passes through points such as \((6, 0)\), \((6, 1)\), \((6, -1)\), and essentially any point where \(x = 6\).
The nature of a vertical line shows that it does not have a slope in the traditional sense—it's often considered undefined—because there is no horizontal change to accompany the vertical one within the line.
Key points about vertical lines include:

• They are parallel to the y-axis.
• All points on the line share the same x-coordinate.
• They do not intersect the y-axis unless that axis represents the same x-coordinate.

These properties make identifying and plotting vertical lines straightforward once you comprehend the influence of x-values on these equations.

Solving Linear Equations

Solving linear equations is a fundamental skill in mathematics, crucial for graphing and understanding a wide range of problems. It involves finding the values of the variables that make the equation true. For any given linear equation, such as \(3x - 18 = 0\), the goal is to isolate the variable, in this case, x.
Here is a simplified step-by-step process:

• Rearrange the equation to isolate terms with the variable on one side. For \(3x - 18 = 0\), add 18 to both sides to have \(3x = 18\).
• Divide each term by the coefficient of the variable, here 3, leading to \(x = \frac{18}{3} = 6\).
• This solution indicates the value of the variable that satisfies the equation.

Understanding these steps helps in quickly solving similar equations and lays the groundwork for more complex algebraic problems. Extracting the value of x shows not only a solution but also visually how the equation behaves when graphed, like forming a vertical line at x = 6 in the current example.