Select The Correct Answer.Which Value Of $x$ Makes The Equation True?$\frac{3x-6}{3} = \frac{7x-3}{6}$A. -11 B. -9 C. -3 D. 3

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Introduction

In mathematics, solving equations is a crucial skill that helps us find the value of unknown variables. In this article, we will focus on solving a linear equation involving fractions. We will break down the solution into manageable steps and provide a clear explanation of each step.

The Equation

The given equation is:

3x−63=7x−36\frac{3x-6}{3} = \frac{7x-3}{6}

Our goal is to find the value of xx that makes this equation true.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 3 and 6 is 6.

\frac{3x-6}{3} \cdot 6 = \frac{7x-3}{6} \cdot 6

This simplifies to:

2(3x−6)=7x−32(3x-6) = 7x-3

Step 2: Distribute the 2

We need to distribute the 2 to both terms inside the parentheses.

2(3x-6) = 6x-12

So, the equation becomes:

6x−12=7x−36x-12 = 7x-3

Step 3: Add 12 to Both Sides

To isolate the term with xx, we need to add 12 to both sides of the equation.

6x-12 + 12 = 7x-3 + 12

This simplifies to:

6x=7x+96x = 7x+9

Step 4: Subtract 7x from Both Sides

We need to subtract 7x from both sides of the equation to get all the xx terms on one side.

6x - 7x = 7x + 9 - 7x

This simplifies to:

−x=9-x = 9

Step 5: Multiply Both Sides by -1

To solve for xx, we need to multiply both sides of the equation by -1.

-x \cdot -1 = 9 \cdot -1

This simplifies to:

x=−9x = -9

Conclusion

We have successfully solved the equation and found the value of xx that makes the equation true. The correct answer is:

A. -9

This solution demonstrates the importance of following the order of operations and simplifying the equation step by step. By breaking down the solution into manageable steps, we can ensure that we arrive at the correct answer.

Tips and Variations

  • When solving equations involving fractions, it's essential to find the least common multiple (LCM) of the denominators and multiply both sides by the LCM.
  • When distributing a coefficient to a term inside parentheses, make sure to multiply the coefficient by each term inside the parentheses.
  • When adding or subtracting the same value to both sides of an equation, the equation remains balanced.

Introduction

In our previous article, we solved a linear equation involving fractions and found the value of xx that makes the equation true. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving equations.

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

A: The first step in solving an equation involving fractions is to find the least common multiple (LCM) of the denominators and multiply both sides of the equation by the LCM.

Q: Why do we need to find the LCM of the denominators?

A: We need to find the LCM of the denominators because it allows us to eliminate the fractions and simplify the equation. By multiplying both sides of the equation by the LCM, we can get rid of the fractions and work with whole numbers.

Q: What is the difference between the distributive property and the distributive law?

A: The distributive property is a rule that states that a single operation can be distributed to multiple terms inside parentheses. The distributive law is a specific application of the distributive property, where a coefficient is multiplied by each term inside parentheses.

Q: How do I know when to add or subtract the same value to both sides of an equation?

A: You know when to add or subtract the same value to both sides of an equation when you want to isolate a term or variable. By adding or subtracting the same value to both sides of the equation, you can keep the equation balanced and ensure that the solution is correct.

Q: What is the order of operations in solving equations?

A: The order of operations in solving equations is:

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

Q: How do I check my solution to an equation?

A: To check your solution to an equation, plug the value of the variable back into the original equation and simplify. If the equation is true, then your solution is correct.

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

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

  • Not following the order of operations
  • Not simplifying the equation correctly
  • Not checking the solution
  • Not considering the domain of the variable

Conclusion

Solving equations can be a challenging task, but with practice and patience, you can become proficient in solving linear equations involving fractions. By following the steps outlined in this Q&A guide, you can ensure that you are solving equations correctly and avoiding common mistakes.

Tips and Variations

  • When solving equations, always follow the order of operations.
  • When simplifying an equation, make sure to combine like terms.
  • When checking your solution, plug the value of the variable back into the original equation.
  • When considering the domain of the variable, make sure to check for any restrictions on the variable.

By following these tips and practicing solving equations, you'll become more confident and proficient in solving linear equations involving fractions.