Solve This Equation. Explain Or Show Your Reasoning.${ \frac{1}{2}x - 7 = \frac{1}{3}(x - 12) }$

Introduction to the Equation

The given equation is a linear equation involving fractions. It is a common type of equation in algebra, and solving it requires careful manipulation of the fractions and variables. The equation is $\frac{1}{2}x - 7 = \frac{1}{3}(x - 12)$. Our goal is to solve for the variable $x$.

Step 1: Multiply both sides by the least common multiple (LCM) of the denominators

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 6. This will allow us to work with whole numbers and simplify the equation.

6 \times \left(\frac{1}{2}x - 7\right) = 6 \times \left(\frac{1}{3}(x - 12)\right)

Step 2: Distribute the multiplication to the terms inside the parentheses

Now that we have multiplied both sides by 6, we can distribute the multiplication to the terms inside the parentheses.

3x - 42 = 2(x - 12)

Step 3: Distribute the multiplication to the terms inside the parentheses again

We need to distribute the multiplication to the terms inside the parentheses again to simplify the equation.

3x - 42 = 2x - 24

Step 4: Add 42 to both sides of the equation

To isolate the variable $x$, we need to add 42 to both sides of the equation.

3x = 2x + 18

Step 5: Subtract 2x from both sides of the equation

Now, we can subtract 2x from both sides of the equation to further isolate the variable $x$.

x = 18

Conclusion

We have successfully solved the equation $\frac{1}{2}x - 7 = \frac{1}{3}(x - 12)$ by multiplying both sides by the least common multiple (LCM) of the denominators, distributing the multiplication, and isolating the variable $x$. The solution to the equation is $x = 18$.

Final Answer

The final answer is $\boxed{18}$.

Discussion

This equation is a simple example of a linear equation involving fractions. Solving it requires careful manipulation of the fractions and variables. The steps involved in solving the equation are:

1. Multiply both sides by the least common multiple (LCM) of the denominators.
2. Distribute the multiplication to the terms inside the parentheses.
3. Add or subtract the same value to both sides of the equation to isolate the variable.
4. Simplify the equation by combining like terms.

These steps can be applied to solve more complex linear equations involving fractions.

Tips and Tricks

When solving linear equations involving fractions, it is essential to:

1. Identify the least common multiple (LCM) of the denominators and multiply both sides by it.
2. Distribute the multiplication to the terms inside the parentheses carefully.
3. Add or subtract the same value to both sides of the equation to isolate the variable.
4. Simplify the equation by combining like terms.

By following these steps and tips, you can solve linear equations involving fractions with ease.

Real-World Applications

Linear equations involving fractions have numerous real-world applications in various fields, such as:

1. Finance: Calculating interest rates and investment returns.
2. Science: Measuring the rate of change of physical quantities.
3. Engineering: Designing and optimizing systems.
4. Economics: Modeling economic systems and predicting outcomes.

These equations are essential in understanding and solving real-world problems.

Common Mistakes

When solving linear equations involving fractions, common mistakes include:

1. Failing to identify the least common multiple (LCM) of the denominators.
2. Not distributing the multiplication to the terms inside the parentheses correctly.
3. Adding or subtracting the wrong value to both sides of the equation.
4. Failing to simplify the equation by combining like terms.

To avoid these mistakes, it is essential to follow the steps and tips outlined above.

Conclusion

In conclusion, solving linear equations involving fractions requires careful manipulation of the fractions and variables. By following the steps and tips outlined above, you can solve these equations with ease. The solution to the equation $\frac{1}{2}x - 7 = \frac{1}{3}(x - 12)$ is $x = 18$.

Introduction

Solving linear equations involving fractions can be a challenging task, but with the right approach and techniques, it can be done with ease. In this article, we will provide a comprehensive Q&A section to help you understand and solve linear equations involving fractions.

Q1: What is the first step in solving a linear equation involving fractions?

A1: The first step in solving a linear equation involving fractions is to identify the least common multiple (LCM) of the denominators and multiply both sides of the equation by it.

Q2: Why is it essential to identify the least common multiple (LCM) of the denominators?

A2: Identifying the least common multiple (LCM) of the denominators is essential because it allows us to eliminate the fractions and work with whole numbers, making it easier to solve the equation.

Q3: How do I distribute the multiplication to the terms inside the parentheses?

A3: To distribute the multiplication to the terms inside the parentheses, you need to multiply each term inside the parentheses by the number outside the parentheses.

Q4: What is the next step after distributing the multiplication?

A4: After distributing the multiplication, the next step is to add or subtract the same value to both sides of the equation to isolate the variable.

Q5: How do I simplify the equation by combining like terms?

A5: To simplify the equation by combining like terms, you need to group the like terms together and add or subtract them.

Q6: What are some common mistakes to avoid when solving linear equations involving fractions?

A6: Some common mistakes to avoid when solving linear equations involving fractions include failing to identify the least common multiple (LCM) of the denominators, not distributing the multiplication to the terms inside the parentheses correctly, adding or subtracting the wrong value to both sides of the equation, and failing to simplify the equation by combining like terms.

Q7: How do I apply the steps to solve a linear equation involving fractions in real-world applications?

A7: To apply the steps to solve a linear equation involving fractions in real-world applications, you need to identify the problem, set up the equation, and then solve it using the steps outlined above.

Q8: What are some real-world applications of linear equations involving fractions?

A8: Some real-world applications of linear equations involving fractions include calculating interest rates and investment returns in finance, measuring the rate of change of physical quantities in science, designing and optimizing systems in engineering, and modeling economic systems and predicting outcomes in economics.

Q9: How do I check my solution to a linear equation involving fractions?

A9: To check your solution to a linear equation involving fractions, you need to plug the solution back into the original equation and verify that it is true.

Q10: What are some tips for solving linear equations involving fractions?

A10: Some tips for solving linear equations involving fractions include being careful when multiplying and dividing fractions, checking your work, and using a calculator to check your solution.

Conclusion

In conclusion, solving linear equations involving fractions requires careful manipulation of the fractions and variables. By following the steps and tips outlined above, you can solve these equations with ease. Remember to identify the least common multiple (LCM) of the denominators, distribute the multiplication to the terms inside the parentheses, add or subtract the same value to both sides of the equation, and simplify the equation by combining like terms. With practice and patience, you will become proficient in solving linear equations involving fractions.

Final Tips

• Always check your work to ensure that your solution is correct.
• Use a calculator to check your solution and verify that it is true.
• Practice solving linear equations involving fractions to become proficient in this skill.
• Apply the steps to solve linear equations involving fractions in real-world applications to see the relevance and importance of this skill.

Resources

• Online resources: Khan Academy, Mathway, and Wolfram Alpha are excellent online resources for learning and practicing linear equations involving fractions.
• Textbooks: Algebra textbooks, such as "Algebra and Trigonometry" by Michael Sullivan, provide comprehensive coverage of linear equations involving fractions.
• Online communities: Join online communities, such as Reddit's r/learnmath and r/math, to connect with other learners and get help with linear equations involving fractions.

Conclusion

Solving linear equations involving fractions is a fundamental skill in algebra and mathematics. By following the steps and tips outlined above, you can solve these equations with ease. Remember to practice regularly and apply the steps to solve linear equations involving fractions in real-world applications to see the relevance and importance of this skill.