Solving (-3 + M) / 9 = 10 A Step-by-Step Guide

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This article provides a detailed walkthrough on how to solve the algebraic equation (-3 + m) / 9 = 10. We will explore the step-by-step process, ensuring a clear understanding of the underlying mathematical principles. This guide is designed for students and anyone looking to enhance their algebra skills. The correct answer is (D) 93. Let's delve into the solution.

Understanding the Equation

In this algebraic equation, our goal is to isolate the variable 'm' to find its value. The equation (-3 + m) / 9 = 10 involves several mathematical operations, including addition, subtraction, multiplication, and division. To solve for 'm', we must reverse these operations in the correct order, maintaining the equation's balance.

The equation states that the expression (-3 + m), which represents a number 'm' with 3 subtracted from it, is divided by 9, resulting in 10. Our task is to find the specific value of 'm' that satisfies this condition. Understanding the structure of the equation is the first step towards finding the solution. We need to undo the division by 9 and then undo the subtraction of 3 to isolate 'm' on one side of the equation. This involves applying inverse operations in the reverse order they appear in the equation. By carefully performing these steps, we can accurately determine the value of 'm' that makes the equation true.

Step-by-Step Solution

To solve the equation (-3 + m) / 9 = 10, we follow these steps:

  1. Isolate the term containing 'm': The first step is to eliminate the division by 9. To do this, we multiply both sides of the equation by 9. This maintains the balance of the equation while isolating the expression containing 'm'.

    (-3 + m) / 9 * 9 = 10 * 9

    This simplifies to:

    -3 + m = 90

    By multiplying both sides by 9, we have successfully removed the denominator and moved closer to isolating 'm'. This is a crucial step in solving algebraic equations, as it simplifies the equation and allows us to focus on the remaining operations.

  2. Isolate 'm': Now, we need to isolate 'm' by eliminating the -3 on the left side of the equation. To do this, we add 3 to both sides of the equation. Again, this keeps the equation balanced while moving us closer to the solution.

    -3 + m + 3 = 90 + 3

    This simplifies to:

    m = 93

    Adding 3 to both sides effectively cancels out the -3, leaving 'm' by itself on the left side. This is the final step in isolating the variable, and it reveals the value of 'm' that satisfies the original equation.

  3. Verify the solution: To ensure our solution is correct, we substitute m = 93 back into the original equation:

    (-3 + 93) / 9 = 10

    90 / 9 = 10

    10 = 10

    Since the equation holds true, our solution is correct. This verification step is essential in algebra to confirm the accuracy of the solution and catch any potential errors in the solving process. By substituting the calculated value back into the original equation, we can be confident that our answer is correct.

Detailed Explanation of Each Step

Step 1: Multiplying Both Sides by 9

The initial equation is (-3 + m) / 9 = 10. To eliminate the fraction, we multiply both sides by 9. This is based on the fundamental principle of algebraic equations: whatever operation you perform on one side, you must perform on the other to maintain equality. By multiplying by 9, we are essentially undoing the division by 9 that is present in the original equation. This step is crucial because it simplifies the equation by removing the denominator, making it easier to work with.

When we multiply the left side, (-3 + m) / 9, by 9, the 9 in the numerator and the 9 in the denominator cancel each other out, leaving us with -3 + m. On the right side, we multiply 10 by 9, resulting in 90. Therefore, after this step, the equation becomes -3 + m = 90. This new equation is much simpler to solve because it no longer involves a fraction. The multiplication step has effectively isolated the expression containing 'm', bringing us closer to finding the value of 'm'.

Step 2: Adding 3 to Both Sides

After multiplying both sides by 9, our equation is -3 + m = 90. To isolate 'm', we need to eliminate the -3 on the left side. We do this by adding 3 to both sides of the equation. Again, this follows the principle of maintaining equality: adding the same value to both sides does not change the solution of the equation. Adding 3 is the inverse operation of subtracting 3, so it effectively cancels out the -3 on the left side.

When we add 3 to the left side, -3 + m + 3, the -3 and +3 cancel each other out, leaving us with just 'm'. On the right side, we add 3 to 90, resulting in 93. Therefore, after this step, the equation becomes m = 93. This step is significant because it directly reveals the value of 'm' that satisfies the equation. By adding 3 to both sides, we have successfully isolated 'm' on one side of the equation, giving us the solution.

Step 3: Verifying the Solution

Once we have found a potential solution, it's essential to verify that it is correct. This is done by substituting the value we found for 'm' back into the original equation and checking if the equation holds true. In this case, we found that m = 93, so we substitute 93 for 'm' in the original equation, (-3 + m) / 9 = 10.

Substituting m = 93, we get (-3 + 93) / 9 = 10. Simplifying the expression inside the parentheses, -3 + 93 equals 90. So the equation becomes 90 / 9 = 10. Dividing 90 by 9 gives us 10, so the equation simplifies to 10 = 10. Since this is a true statement, it confirms that our solution, m = 93, is correct. Verification is a crucial step in solving equations because it helps us catch any mistakes we may have made during the solving process. By substituting the solution back into the original equation, we can be confident that our answer is accurate.

Why Option D is Correct

As we have demonstrated through the step-by-step solution, the correct value for 'm' is 93. This corresponds to option D in the given choices. Options A, B, and C are incorrect because they do not satisfy the original equation when substituted for 'm'. By carefully following the steps of isolating 'm', we have arrived at the accurate answer. Option D is the only choice that makes the equation (-3 + m) / 9 = 10 true.

Common Mistakes to Avoid

When solving algebraic equations, several common mistakes can lead to incorrect answers. It is important to be aware of these mistakes and take steps to avoid them.

One common mistake is failing to perform the same operation on both sides of the equation. Remember, the golden rule of algebra is that whatever you do to one side of the equation, you must do to the other to maintain equality. For example, if you multiply one side by a number, you must multiply the other side by the same number. Neglecting to do so will result in an unbalanced equation and an incorrect solution.

Another frequent error is incorrectly applying the order of operations. When solving equations, it is essential to follow the reverse order of operations (PEMDAS/BODMAS). This means addressing addition and subtraction before multiplication and division. Failing to follow the correct order can lead to solving the equation in the wrong sequence and arriving at an incorrect answer.

Sign errors are also common mistakes. Be careful when dealing with negative numbers and ensure you apply the correct rules of arithmetic. For instance, subtracting a negative number is the same as adding a positive number, and multiplying or dividing two negative numbers results in a positive number. Overlooking these rules can lead to sign errors and an incorrect solution.

Finally, not verifying the solution is a significant mistake. Always substitute your solution back into the original equation to check if it holds true. This step helps catch any errors you may have made during the solving process and ensures your answer is accurate.

Tips for Solving Algebraic Equations

To improve your ability to solve algebraic equations, consider the following tips:

  1. Understand the Basics: Make sure you have a solid understanding of the fundamental concepts of algebra, such as variables, constants, coefficients, and operations. A strong foundation is essential for tackling more complex equations.
  2. Follow the Order of Operations: Remember the order of operations (PEMDAS/BODMAS) and apply it consistently. This will help you solve equations in the correct sequence and avoid errors.
  3. Isolate the Variable: The primary goal in solving an equation is to isolate the variable. Use inverse operations to undo the operations that are being performed on the variable. For example, if the variable is being multiplied by a number, divide both sides of the equation by that number.
  4. Keep the Equation Balanced: Always perform the same operation on both sides of the equation to maintain equality. This ensures that the equation remains balanced and the solution remains correct.
  5. Simplify Expressions: Before solving, simplify any expressions on either side of the equation. This may involve combining like terms, distributing, or factoring.
  6. Check Your Work: After solving an equation, always substitute your solution back into the original equation to verify that it is correct. This will help you catch any errors you may have made.
  7. Practice Regularly: The best way to improve your algebra skills is to practice regularly. Solve a variety of equations to build your confidence and proficiency.

By following these tips, you can enhance your problem-solving abilities and approach algebraic equations with greater confidence.

Conclusion

In conclusion, solving the equation (-3 + m) / 9 = 10 requires a systematic approach that involves isolating the variable 'm' by performing inverse operations. The correct solution is m = 93, which corresponds to option D. By understanding the steps involved and practicing regularly, you can improve your algebra skills and confidently solve similar equations. Remember to always verify your solution to ensure accuracy. This comprehensive guide has provided a thorough explanation of the solution process, common mistakes to avoid, and helpful tips for solving algebraic equations. With this knowledge, you can approach algebraic problems with greater confidence and achieve success.