Solving Linear Equations: A Comprehensive Guide to Understanding and Solving the Equation 3z912

Linear equations are a fundamental aspect of algebra, and mastering them is crucial for advancing to more complex mathematical concepts. In this article, we will delve into the steps to solve the linear equation 3z912, providing a step-by-step guide for clarity and understanding.

Understanding Linear Equations

A linear equation is an algebraic equation where the variable (or variables) is raised to the first power. The general form of a linear equation is ax b c, where a, b, and c are constants, and x is the variable. In our specific equation, 3z912, the variable is z and 9 is a constant, but it can be simplified and solved effectively.

Step-by-Step Guide to Solving 3z912

Let's break down the process of solving the equation step by step:

Step 1: Simplify the Equation

The given equation is 3z912. First, we need to isolate the term with the variable z. Notice that the 9 is a separate term, so we can subtract 9 from both sides to simplify:

3z9 – 9 12 – 9

This simplifies to:

3z 3

Step 2: Isolate the Variable

Next, we need to isolate the variable z. To do this, we divide both sides of the equation by 3:

3z/3 3/3

This simplifies to:

z 1

Step 3: Verify the Solution

To ensure our solution is correct, we can substitute z 1 back into the original equation and check if it holds true:

3(1)9 12

This simplifies to:

39 12

After removing the parentheses, we see that:

3 - 9 12 - 9

Which simplifies to:

3 3

The equation holds true, confirming that z 1 is the correct solution.

Alternative Method

Another way to solve this equation involves moving the constants to the other side. Let's take a look at this method:

Method 1: Moving Constants

Starting with 3z912, we can move the 9 to the left side by subtracting it from 12:

3z 12 – 9

This simplifies to:

3z 3

Then, we divide both sides by 3 to isolate z:

z 3/3

This simplifies to:

z 1

Method 2: Simplifying Directly

We can simplify the equation directly by recognizing that 3z9 can be interpreted as 3z - 9:

3z - 9 12

Next, we add 9 to both sides to isolate the term with z:

3z 12 9

This simplifies to:

3z 21

Finally, we divide both sides by 3 to solve for z:

z 21/3

This simplifies to:

z 7

However, this method does not correctly solve the original equation, as it changes the form of the equation. It is important to stick to the correct method of simplification as shown in the initial steps.

Conclusion

Solving the equation 3z912 involves basic algebraic manipulation and understanding. By following the steps outlined, one can systematically solve such equations. The key steps include simplifying the equation to isolate the variable, and then verifying the solution.

For further practice and understanding, it is recommended to solve similar equations and explore other linear equations. Understanding the underlying principles will provide a strong foundation for tackling more complex mathematical concepts in the future.