Problem 61 - Introductory Algebra for College Students

Question

Graph equation. \(12-3 x=0\)

Step-by-Step Solution

Answer

The graph is a vertical line passing through the point (4,0).

Step 1: Rearrange equation

Rearrange the equation to solve for \(x\). The equation now is \(12 - 3x = 0\). Add 3x to both sides to get \(12 = 3x\). Then, divide both sides by 3 to isolate \(x\), the equation will then be \(x = 12/3\).

Step 2: Solve for x

Now solve for \(x\) by performing the division operation from the equation \(x = 12/3\), which gives \(x = 4\).

Step 3: Plot the value of x on a graph

Plotting this value on a graph would yield a vertical line at \(x = 4\). Since this formula is in the form \(x = a\), where \(a\) is a constant, the graph would be a vertical line passing through the point (4,0).

Key Concepts

Graphing Equations

Graphing equations can be an exciting way to visually understand solutions, especially for students learning algebra. When you graph an equation like the one in our exercise, you are essentially representing a set of solutions on a coordinate plane. In the equation we discussed, after finding the solution for \(x\), we plot the result on a graph.

For the equation \(12-3x=0\), once it's solved, we know \(x = 4\). This means that on a 2-dimensional graph, we will draw a straight, vertical line at \(x = 4\).

• The graph \(x = a\) (where \(a\) is a constant) always represents a vertical line.
• A vertical line indicates that \(x\) has the same value for every possible \(y\) value.

This is important because the graph gives a clear, visual representation of all solutions to the equation in "one picture".

Rearranging Equations

Rearranging equations is a crucial skill in algebra. It involves manipulating the equation to isolate the desired variable. In the case of the given exercise, our original equation is \(12 - 3x = 0\).

To rearrange this equation, we need to perform operations that will help us solve for \(x\).

• Firstly, add \(3x\) to both sides, making it \(12 = 3x\).
• Next, divide both sides by 3 to get \(x\) alone, resulting in \(x = 4\).

By keeping the equation balanced, you ensure that the equality holds true, which is a fundamental principle when rearranging equations.

Each rearrangement step should move you closer to isolating \(x\), ensuring clarity and accuracy in your solution.

Solving for x

Solving for \(x\) is the process of determining what numerical value \(x\) must be to satisfy the equation. It's a common technique in algebra aimed at uncovering unknowns. In our exercise, we've already rearranged the equation to \(12 = 3x\).

From here, solving for \(x\) requires performing a basic arithmetic operation.

• Divide each side by 3: \(x = \frac{12}{3}\).
• This yields \(x = 4\).

This step involves clear thinking and careful arithmetic to ensure accuracy.

Once you reach this step, always double-check your work to confirm that \(x\) indeed satisfies the original equation. This ensures you have solved correctly and understand the relationship between the components of the equation.