Solving the Equation 3x 2 14 and Finding the Value of x

Algebra is a fundamental branch of mathematics that deals with the study of symbols and the rules for manipulating these symbols. Solving equations is one of the core skills in algebra. In this article, we will walk through a solution to the equation 3x 2 14, step by step, to find the value of x. This process will help you understand the underlying principles and techniques in algebra that are essential for solving similar equations in the future.

Step-by-Step Solution

The equation given is 3x 2 14. This is a linear equation, which involves a variable (x) and a constant, and the variable is not raised to any power other than 1. Let's solve this equation step by step.

Step 1: Isolate the Variable Term

The first step in solving any algebraic equation is to isolate the term that contains the variable (x) on one side of the equation. We start by subtracting 2 from both sides of the equation:

3x 2 14 3x 2 - 2 14 - 2 3x 12

Step 2: Solve for the Variable

Now that we have isolated the term containing the variable, the next step is to solve for the variable. To do this, we divide both sides of the equation by 3 (the coefficient of x):

3x 12 (frac{3x}{3} frac{12}{3}) x 4

Thus, the value of x is 4.

Understanding the Process

Let's break down the steps in more detail to ensure you fully understand the process:

Subtraction Step

First, we subtract 2 from both sides of the equation 3x 2 14:

3x 2 - 2 14 - 2

This simplifies to:

3x 12

We subtracted the same number from both sides to keep the equation balanced. Subtracting the same value from both sides of an equation helps maintain the equality.

Division Step

Next, we divide both sides of the equation 3x 12 by 3:

(frac{3x}{3} frac{12}{3})

This simplifies to:

x 4

We divided by the coefficient of x (which is 3) to isolate x. Dividing both sides of an equation by the same non-zero number maintains the equality.

Common Questions and Clarifications

Q: Can the value of x be negative?

No, based on the given equation and the steps taken to solve it, the value of x is 4. If the equation were to change, the value of x could be negative, positive, or any other real number, depending on the values of the constants and coefficients in the equation.

Q: What if the constant on the left side is not given as 2, but as another constant?

The process would be similar. You would subtract the given constant from both sides, then divide both sides by the coefficient of x. For example, if the equation was 3x 5 19, you would first subtract 5 from both sides to get 3x 14, then divide both sides by 3 to get x (frac{14}{3}).

Practice Problems

1. Solve the equation 2x 3 11 to find the value of x.

2. Solve the equation 4x - 5 11 to find the value of x.

3. Verify your solution by plugging the value of x back into the original equation to ensure it balances.

Conclusion

In this article, we solved the equation 3x 2 14 step by step to determine that the value of x is 4. The process involved isolating the variable term and then solving for the variable. Understanding these steps and practicing similar problems will help you develop a solid foundation in algebra and improve your problem-solving skills.