Problem 57 - College Algebra

Question

Graph each 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 passing through \(x = 6\) on a rectangular coordinate system.

Step 1: Simplify the Equation

Firstly, it is needed to simplify the equation. The given equation is \(3x - 18 = 0\). In order to isolate \(x\), add 18 to both sides of the equation which gives \(3x = 18\).

Step 2: Solve for x

Next, solve for \(x\) by dividing both sides of the equation by 3 to get \(x = 6\).

Step 3: Graphing

On a rectangular coordinate system, mark the point where \(x = 6\) on the x-axis as this equation represents a vertical line passing through \(x = 6\). Draw a vertical line through this point to graph the given equation.

Key Concepts

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, is a two-dimensional plane made up of two perpendicular lines called axes. These are the x-axis, which runs horizontally, and the y-axis, which runs vertically. Intersecting at the origin point \(0,0\), these axes divide the plane into four quadrants.
You can use this system to plot points and graph equations quite easily. Each point on this plane is defined by an ordered pair \(x, y\), where \(x\) and \(y\) are the respective distances from the y-axis and x-axis. These coordinates help in pinpointing exact locations on the grid.
Graphing equations using this system involves plotting each point that satisfies the equation. Then you can connect these points to form a line or a curve, depending on the type of equation you're working with.

Vertical Line

A vertical line in the rectangular coordinate system is a line where all points on the line have the same x-coordinate. This implies that no matter the value of \(y\), \(x\) remains constant. In other words, a vertical line is parallel to the y-axis.
For example, the equation \(x = 6\) represents a vertical line. You don't need to change or solve for \(y\) because \(y\) can be any value. You only need to mark \(x = 6\) on the x-axis and draw a line that extends vertically through this point. Vertical lines highlight a constant x-value across all possible y-values.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Linear equations are characterized by their constant rate of change and are usually in the form of \text{Ax} + \text{B} = 0\ where \text{A}\ and \text{B}\ are constants.
To solve the equation \(3x - 18 = 0\), you'll first want to isolate the variable \(x\). This is done by adding 18 to both sides, resulting in \(3x = 18\).
Next, divide every term by 3 to solve for \(x\), giving you \(x = 6\). The solution tells you that when \(x\) is 6, the equation holds true. This step-by-step approach to solving for \(x\) ensures all actions are consistent and valid across both sides of the equation. Now, it's easier to graph on the coordinate plane, as you know exactly where the line should be plotted.