Which Procedure Can Be Used To Solve The Equation $540 = 12z$, And What Is The Solution?A. Multiply Both Sides By 12; The Solution Is 6,480.B. Multiply $12z$ By 12; The Solution Is 540.C. Divide Both Sides By 12; The Solution Is 45.D.

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a simple linear equation, $540 = 12z$, and explore the different procedures that can be used to find the solution.

Understanding the Equation

The given equation is $540 = 12z$. This is a linear equation in one variable, where $z$ is the variable. The equation states that the product of $12$ and $z$ is equal to $540$.

Procedure A: Multiply Both Sides by 12

One possible procedure to solve the equation is to multiply both sides by $12$. This is a common technique used to isolate the variable.

• Step 1: Multiply both sides of the equation by $12$.
• Step 2: Simplify the equation to isolate the variable.

Using this procedure, we get:

$$12 \times 540 = 12 \times 12z$$

$$6480 = 144z$$

Now, we can solve for $z$ by dividing both sides by $144$.

$$z = \frac{6480}{144}$$

$$z = 45$$

Therefore, the solution to the equation $540 = 12z$ using procedure A is $z = 45$.

Procedure B: Multiply $12z$ by 12

Another possible procedure to solve the equation is to multiply $12z$ by $12$. This will also help us isolate the variable.

• Step 1: Multiply $12z$ by $12$.
• Step 2: Simplify the equation to isolate the variable.

Using this procedure, we get:

$$12 \times 12z = 12 \times 540$$

$$144z = 6480$$

Now, we can solve for $z$ by dividing both sides by $144$.

$$z = \frac{6480}{144}$$

$$z = 45$$

Therefore, the solution to the equation $540 = 12z$ using procedure B is also $z = 45$.

Procedure C: Divide Both Sides by 12

A third possible procedure to solve the equation is to divide both sides by $12$. This will also help us isolate the variable.

• Step 1: Divide both sides of the equation by $12$.
• Step 2: Simplify the equation to isolate the variable.

Using this procedure, we get:

$$\frac{540}{12} = z$$

$$45 = z$$

Therefore, the solution to the equation $540 = 12z$ using procedure C is $z = 45$.

Conclusion

In this article, we explored three different procedures to solve the linear equation $540 = 12z$. We found that all three procedures led to the same solution, $z = 45$. This demonstrates the importance of understanding the different techniques used to solve linear equations and the need to verify the solution using multiple methods.

Key Takeaways

• Linear equations are a fundamental concept in mathematics.
• Solving linear equations requires a clear understanding of the different procedures used to isolate the variable.
• Verifying the solution using multiple methods is essential to ensure accuracy.

Frequently Asked Questions

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1.

Q: What is the solution to the equation $540 = 12z$?

A: The solution to the equation $540 = 12z$ is $z = 45$.

Q: What are the different procedures used to solve linear equations?

A: The different procedures used to solve linear equations include multiplying both sides by a constant, dividing both sides by a constant, and using inverse operations.

Q: Why is it essential to verify the solution using multiple methods?

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It can be written in the form of $ax = b$, where $a$ and $b$ are constants, and $x$ is the variable.

Q: What is the solution to the equation $540 = 12z$?

A: The solution to the equation $540 = 12z$ is $z = 45$. This can be found by multiplying both sides of the equation by $12$, dividing both sides by $12$, or using inverse operations.

Q: What are the different procedures used to solve linear equations?

A: The different procedures used to solve linear equations include:

• Multiplying both sides by a constant: This involves multiplying both sides of the equation by a constant to isolate the variable.
• Dividing both sides by a constant: This involves dividing both sides of the equation by a constant to isolate the variable.
• Using inverse operations: This involves using inverse operations, such as addition and subtraction, to isolate the variable.

Q: Why is it essential to verify the solution using multiple methods?

A: Verifying the solution using multiple methods ensures accuracy and helps to build confidence in the solution. It also helps to identify any errors or inconsistencies in the solution.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

• Not following the order of operations: This can lead to incorrect solutions.
• Not checking for extraneous solutions: This can lead to incorrect solutions.
• Not using inverse operations correctly: This can lead to incorrect solutions.

Q: How can I check if my solution is correct?

A: To check if your solution is correct, you can:

• Plug the solution back into the original equation: If the solution is correct, the equation should be true.
• Use multiple methods to solve the equation: If the solution is correct, it should be the same using multiple methods.
• Check for extraneous solutions: If the solution is correct, it should not be an extraneous solution.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

• Finance: Linear equations are used to calculate interest rates, investments, and loans.
• Science: Linear equations are used to model population growth, chemical reactions, and physical systems.
• Engineering: Linear equations are used to design and optimize systems, such as bridges and buildings.

Q: How can I practice solving linear equations?

A: To practice solving linear equations, you can:

• Use online resources: There are many online resources available that provide practice problems and exercises.
• Work with a tutor or teacher: A tutor or teacher can provide personalized guidance and feedback.
• Practice with real-world examples: Practice solving linear equations using real-world examples and applications.

Conclusion

Solving linear equations is an essential skill in mathematics, and it has many real-world applications. By understanding the different procedures used to solve linear equations and verifying the solution using multiple methods, you can build confidence in your solution and ensure accuracy.