Solving For X In The Equation (2/3)((1/2)x + 12) = (1/2)((1/3)x + 14) - 3

In this comprehensive guide, we will delve into the process of solving for the unknown variable x in the given equation:

$$\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3$$

This equation involves fractions, parentheses, and constants, making it a classic example of a linear equation. Our goal is to isolate x on one side of the equation to determine its value. To achieve this, we will employ a series of algebraic manipulations, ensuring that each step maintains the equality of the equation. This step-by-step approach will not only lead us to the solution but also enhance your understanding of the fundamental principles of equation solving.

1. Distribute the Constants

The first step in simplifying the equation is to distribute the constants outside the parentheses to the terms inside. This involves multiplying ${\frac{2}{3}}$ with both ${\frac{1}{2}x}$ and 12 on the left side, and multiplying ${\frac{1}{2}}$ with both ${\frac{1}{3}x}$ and 14 on the right side. This process will eliminate the parentheses and pave the way for combining like terms.

Detailed Breakdown

On the left side, we have:

$$\frac{2}{3} \times \frac{1}{2}x = \frac{2}{6}x = \frac{1}{3}x$$

$$\frac{2}{3} \times 12 = \frac{24}{3} = 8$$

So, the left side of the equation becomes ${\frac{1}{3}x + 8}$.

On the right side, we have:

$$\frac{1}{2} \times \frac{1}{3}x = \frac{1}{6}x$$

$$\frac{1}{2} \times 14 = 7$$

So, the right side of the equation becomes ${\frac{1}{6}x + 7}$. Remember to subtract 3 from this result, giving us ${\frac{1}{6}x + 7 - 3 = \frac{1}{6}x + 4}$.

After distributing the constants, our equation now looks like this:

$$\frac{1}{3}x + 8 = \frac{1}{6}x + 4$$

This simplified form allows us to proceed with the next step, which involves gathering the x terms on one side and the constant terms on the other.

2. Combine Like Terms

Now that we've eliminated the parentheses through distribution, the next crucial step is to combine like terms. This involves rearranging the equation so that all terms containing x are on one side and all constant terms are on the other. This strategic regrouping simplifies the equation and brings us closer to isolating x. By performing this step carefully, we maintain the balance of the equation while making it easier to solve.

Isolating x Terms

To begin, let's focus on moving the x terms to one side of the equation. We have ${\frac{1}{3}x}$ on the left side and ${\frac{1}{6}x}$ on the right side. To eliminate the ${\frac{1}{6}x}$ term from the right side, we subtract ${\frac{1}{6}x}$ from both sides of the equation. This ensures that the equation remains balanced.

$$\frac{1}{3}x + 8 - \frac{1}{6}x = \frac{1}{6}x + 4 - \frac{1}{6}x$$

This simplifies to:

$$\frac{1}{3}x - \frac{1}{6}x + 8 = 4$$

Now, we need to combine the x terms. To do this, we find a common denominator for ${\frac{1}{3}}$ and ${\frac{1}{6}}$, which is 6. We rewrite ${\frac{1}{3}x}$ as ${\frac{2}{6}x}$, so the equation becomes:

$$\frac{2}{6}x - \frac{1}{6}x + 8 = 4$$

Combining the x terms, we get:

$$\frac{1}{6}x + 8 = 4$$

Isolating Constant Terms

Next, we need to move the constant terms to the other side of the equation. We have +8 on the left side, so we subtract 8 from both sides to eliminate it from the left side:

$$\frac{1}{6}x + 8 - 8 = 4 - 8$$

This simplifies to:

$$\frac{1}{6}x = -4$$

Now, the equation is in a much simpler form, with the x term isolated on one side and a constant on the other. This prepares us for the final step of solving for x.

3. Isolate x

With the equation now simplified to ${\frac{1}{6}x = -4}$, the final step is to isolate x completely. This involves eliminating the coefficient ${\frac{1}{6}}$ that is multiplying x. To do this, we perform the inverse operation, which is multiplication by the reciprocal of the coefficient. This strategic manipulation will reveal the value of x and provide the solution to the equation.

Multiplying by the Reciprocal

The coefficient of x is ${\frac{1}{6}}$, and its reciprocal is 6. To eliminate the coefficient, we multiply both sides of the equation by 6. This maintains the balance of the equation while isolating x.

$$6 \times \frac{1}{6}x = 6 \times (-4)$$

On the left side, the multiplication simplifies to:

$$6 \times \frac{1}{6}x = 1x = x$$

On the right side, the multiplication yields:

$$6 \times (-4) = -24$$

Therefore, the equation now becomes:

$$x = -24$$

This result tells us that the value of x that satisfies the original equation is -24. We have successfully isolated x and determined its value through a series of algebraic manipulations.

4. Verify the Solution

To ensure the accuracy of our solution, it's crucial to verify it by substituting the value we found for x back into the original equation. This process confirms that our solution satisfies the equation and that no errors were made during the solving process. By performing this verification, we can have confidence in the correctness of our answer.

Substituting x = -24

Our original equation is:

$$\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3$$

We substitute x = -24 into this equation:

$$\frac{2}{3}\left(\frac{1}{2} (-24)+12\right)=\frac{1}{2}\left(\frac{1}{3} (-24)+14\right)-3$$

Now, we simplify both sides of the equation separately.

Simplifying the Left Side

First, we evaluate the expression inside the parentheses:

$$\frac{1}{2} (-24) = -12$$

So, the expression inside the parentheses becomes:

$$-12 + 12 = 0$$

Now, we multiply by ${\frac{2}{3}}$:

$$\frac{2}{3} \times 0 = 0$$

Thus, the left side of the equation simplifies to 0.

Simplifying the Right Side

First, we evaluate the expression inside the parentheses:

$$\frac{1}{3} (-24) = -8$$

So, the expression inside the parentheses becomes:

$$-8 + 14 = 6$$

Now, we multiply by ${\frac{1}{2}}$:

$$\frac{1}{2} \times 6 = 3$$

Finally, we subtract 3:

$$3 - 3 = 0$$

Thus, the right side of the equation also simplifies to 0.

Conclusion of Verification

Since both sides of the equation simplify to 0 when x = -24, our solution is verified. This confirms that x = -24 is indeed the correct solution to the equation.

Conclusion

In summary, we have successfully solved for x in the equation ${\frac{2}{3}\left(\frac{1}{2} x+12\right)=\frac{1}{2}\left(\frac{1}{3} x+14\right)-3}$. Through a systematic approach involving distribution, combining like terms, and isolating x, we determined that the value of x is -24. We further verified this solution by substituting it back into the original equation, confirming its accuracy. This step-by-step guide not only provides the solution but also illustrates the fundamental principles of equation solving in algebra.

Key Takeaways

• Distribution: The first step to simplify equations with parentheses. Distribute constant terms to terms inside the parenthesis.
• Combining Like Terms: Collect and combine similar terms by adding/subtracting constants and x variables on both sides.
• Isolating the Variable: Use inverse operations to isolate the variable you're solving for.
• Verification: Substitute the solution back into the original equation to verify correctness.

Understanding these steps is crucial for solving various algebraic equations and is a fundamental skill in mathematics.