Incorrectly simplifying the equation

Oct 13, 2025 by miodeya.com 30 views

Introduction

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we will explore the steps taken by Paloma to solve the equation $6x + 5 = 3(2x + 1)$ and understand the meaning of her result, $2 = 0$ . We will break down the steps involved in solving the equation and provide a clear explanation of the final result.

Step 1: Distributing the 3

The first step in solving the equation is to distribute the 3 to the terms inside the parentheses. This involves multiplying the 3 by each term inside the parentheses.

6x + 5 = 3(2x + 1)
6x + 5 = 6x + 3

Step 2: Simplifying the Equation

After distributing the 3, the equation becomes $6x + 5 = 6x + 3$ . The next step is to simplify the equation by combining like terms.

6x + 5 = 6x + 3

Step 3: Subtracting 6x from Both Sides

To isolate the constant term, we need to subtract $6x$ from both sides of the equation.

6x + 5 - 6x = 6x + 3 - 6x
5 = 3

Step 4: Understanding the Result

The final result of Paloma's equation is $2 = 0$ . However, this result is not a valid solution to the equation. In fact, it is a contradiction, as 2 cannot be equal to 0.

What Went Wrong?

So, what went wrong in Paloma's solution? The problem lies in the fact that she incorrectly simplified the equation. By subtracting $6x$ from both sides, she eliminated the variable $x$ , which is not allowed in algebra.

The Correct Solution

To solve the equation correctly, we need to follow the order of operations and simplify the equation step by step.

6x + 5 = 3(2x + 1)
6x + 5 = 6x + 3

Subtracting $6x$ from both sides gives us:

5 = 3

However, this is still not a valid solution, as 5 cannot be equal to 3. The correct solution involves recognizing that the equation is a contradiction, and there is no valid solution.

Conclusion

In conclusion, Paloma's result, $2 = 0$ , is not a valid solution to the equation $6x + 5 = 3(2x + 1)$ . The correct solution involves recognizing that the equation is a contradiction, and there is no valid solution. By following the order of operations and simplifying the equation step by step, we can arrive at the correct conclusion.

Common Mistakes to Avoid

When solving equations, it is essential to avoid common mistakes that can lead to incorrect solutions. Some common mistakes include:

Incorrectly simplifying the equation: This can lead to the elimination of variables, which is not allowed in algebra.
Not following the order of operations: This can result in incorrect solutions and contradictions.
Not recognizing contradictions: This can lead to incorrect solutions and a lack of understanding of the equation.

Tips for Solving Equations

To solve equations correctly, follow these tips:

Follow the order of operations: This involves simplifying the equation step by step, using the correct order of operations.
Recognize contradictions: If the equation is a contradiction, there is no valid solution.
Avoid incorrectly simplifying the equation: This can lead to the elimination of variables, which is not allowed in algebra.
Use algebraic properties: This involves using properties such as the distributive property and the commutative property to simplify the equation.

Conclusion

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to simplify the equation by combining like terms and using algebraic properties.

Q: How do I distribute a coefficient to terms inside parentheses?

A: To distribute a coefficient to terms inside parentheses, you multiply the coefficient by each term inside the parentheses.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when simplifying an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I recognize a contradiction in an equation?

A: A contradiction in an equation occurs when the equation is false, such as 2 = 0. To recognize a contradiction, look for any statements that are clearly false.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Q: How do I solve an equation with variables on both sides?

A: To solve an equation with variables on both sides, you need to isolate the variable by performing the same operation on both sides of the equation.

Q: What is the distributive property?

A: The distributive property is a property of algebra that states that a coefficient can be distributed to terms inside parentheses.

Q: How do I use the distributive property to simplify an equation?

A: To use the distributive property to simplify an equation, multiply the coefficient by each term inside the parentheses.

Q: What is the commutative property?

A: The commutative property is a property of algebra that states that the order of the terms in an expression does not change the value of the expression.

Q: How do I use the commutative property to simplify an equation?

A: To use the commutative property to simplify an equation, rearrange the terms in the equation to make it easier to solve.

Q: What is the associative property?

A: The associative property is a property of algebra that states that the order in which you perform operations does not change the value of the expression.

Q: How do I use the associative property to simplify an equation?

A: To use the associative property to simplify an equation, rearrange the terms in the equation to make it easier to solve.

Q: What is the identity property?

A: The identity property is a property of algebra that states that a value does not change the value of an expression.

Q: How do I use the identity property to simplify an equation?

A: To use the identity property to simplify an equation, add or subtract zero to the equation to make it easier to solve.

Conclusion

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. By following the order of operations and simplifying the equation step by step, we can arrive at the correct conclusion. However, it is essential to avoid common mistakes that can lead to incorrect solutions, such as incorrectly simplifying the equation and not recognizing contradictions. By following the tips outlined in this article, we can solve equations correctly and arrive at the correct conclusion.