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

Introduction

In mathematics, solving equations is a fundamental concept that helps us find the value of unknown variables. In this article, we will focus on solving a specific equation, $\frac{3x-6}{3}=\frac{7x-3}{6}$, and determine which value of $x$ makes the equation true. We will use algebraic techniques to simplify the equation and isolate the variable $x$.

Understanding the Equation

The given equation is a rational equation, which means it contains fractions with variables in the numerator and denominator. To solve this equation, we need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Step 1: Multiply Both Sides by the LCM

The LCM of 3 and 6 is 6. We will multiply both sides of the equation by 6 to eliminate the fractions.

$$6 \times \frac{3x-6}{3} = 6 \times \frac{7x-3}{6}$$

Step 2: Simplify the Equation

After multiplying both sides by 6, we can simplify the equation by canceling out the common factors.

$$2(3x-6) = 7x-3$$

Step 3: Distribute the 2

We will distribute the 2 to the terms inside the parentheses.

$$6x-12 = 7x-3$$

Step 4: Isolate the Variable $x$

To isolate the variable $x$, we need to get all the terms with $x$ on one side of the equation. We can do this by subtracting $6x$ from both sides of the equation.

$$-12 = x-3$$

Step 5: Add 3 to Both Sides

We will add 3 to both sides of the equation to isolate the variable $x$.

$$-9 = x$$

Conclusion

After solving the equation, we found that the value of $x$ that makes the equation true is $-9$. This is the correct answer, and we can verify it by plugging $x=-9$ back into the original equation.

Verification

Let's plug $x=-9$ back into the original equation to verify our answer.

$$\frac{3(-9)-6}{3}=\frac{7(-9)-3}{6}$$

$$\frac{-27-6}{3}=\frac{-63-3}{6}$$

$$\frac{-33}{3}=\frac{-66}{6}$$

$$-11=-11$$

As we can see, the equation is true when $x=-9$. Therefore, the correct answer is B. -9.

Final Answer

The final answer is B. -9.

Introduction

In our previous article, we solved the equation $\frac{3x-6}{3}=\frac{7x-3}{6}$ and found that the value of $x$ that makes the equation true is $-9$. In this article, we will provide a Q&A section to help clarify any doubts and provide additional information on solving rational equations.

Q&A

Q: What is a rational equation?

A: A rational equation is an equation that contains fractions with variables in the numerator and denominator.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the least common multiple (LCM)?

A: The LCM is the smallest multiple that is common to both numbers. For example, the LCM of 3 and 6 is 6.

Q: How do I find the LCM?

A: To find the LCM, you can list the multiples of each number and find the smallest multiple that is common to both. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

Q: What is the greatest common divisor (GCD)?

A: The GCD is the largest number that divides both numbers without leaving a remainder.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you need to cancel out any common factors between the numerator and denominator.

Q: What is the difference between a rational equation and a rational expression?

A: A rational equation is an equation that contains fractions with variables in the numerator and denominator, while a rational expression is an expression that contains fractions with variables in the numerator and denominator.

Q: Can I use the same steps to solve a rational expression?

A: Yes, you can use the same steps to solve a rational expression as you would to solve a rational equation.

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

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

• Not eliminating the fractions by multiplying both sides of the equation by the LCM
• Not simplifying the equation by canceling out common factors
• Not isolating the variable on one side of the equation

Q: How do I verify my answer?

A: To verify your answer, you can plug the value of the variable back into the original equation and check if the equation is true.

Conclusion

Solving rational equations can be challenging, but with practice and patience, you can master the skills needed to solve these types of equations. Remember to eliminate the fractions by multiplying both sides of the equation by the LCM, simplify the equation by canceling out common factors, and isolate the variable on one side of the equation. By following these steps and avoiding common mistakes, you can solve rational equations with confidence.

Final Tips

• Always start by eliminating the fractions by multiplying both sides of the equation by the LCM.
• Simplify the equation by canceling out common factors.
• Isolate the variable on one side of the equation.
• Verify your answer by plugging the value of the variable back into the original equation.

By following these tips and practicing regularly, you can become proficient in solving rational equations and tackle more complex math problems with confidence.