Solving For M: A Step-by-Step Guide

Solving algebraic equations is a fundamental skill in mathematics. In this article, we will walk you through a step-by-step process to solve the equation 5(m+1) = -2(m+8). Whether you're a student tackling homework or just looking to brush up on your algebra skills, this guide will provide a clear and concise explanation. We'll break down each step, ensuring you understand the logic behind the solution. Let's dive in and unravel this equation together!

Understanding the Basics of Algebraic Equations

Before we jump into solving the equation, it’s important to understand the basic principles of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions involve variables (like 'm' in our case) and constants, connected by mathematical operations. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. This involves isolating the variable on one side of the equation by performing the same operations on both sides to maintain the balance.

In our specific equation, 5(m+1) = -2(m+8), we have a linear equation in one variable, 'm'. Linear equations are characterized by the variable being raised to the power of 1. The steps to solve such equations typically involve distributing, combining like terms, and isolating the variable. Understanding these fundamental concepts is crucial for successfully tackling more complex algebraic problems in the future. Remember, the key is to maintain balance and systematically simplify the equation until you can determine the value of the unknown variable.

Step-by-Step Solution to 5(m+1) = -2(m+8)

Let's break down the solution to the equation 5(m+1) = -2(m+8) into manageable steps.

Step 1: Distribute

The first step in solving this equation is to eliminate the parentheses by distributing the numbers outside them. This involves multiplying the number outside the parenthesis by each term inside. So, on the left side, we multiply 5 by both 'm' and '1', and on the right side, we multiply -2 by both 'm' and '8'. This gives us:

5 * m + 5 * 1 = -2 * m + (-2) * 8

Simplifying this, we get:

5m + 5 = -2m - 16

This step is crucial because it clears the way for combining like terms and isolating the variable 'm'. Distributive property is a cornerstone of algebraic manipulation, and mastering this step will greatly enhance your ability to solve equations.

Step 2: Combine Like Terms

The next step is to gather all the terms containing 'm' on one side of the equation and the constant terms on the other side. To do this, we'll add 2m to both sides of the equation to eliminate the '-2m' term on the right side. This ensures that all 'm' terms are on the left.

5m + 5 + 2m = -2m - 16 + 2m

This simplifies to:

7m + 5 = -16

Now, we need to move the constant term (+5) from the left side to the right side. We do this by subtracting 5 from both sides of the equation:

7m + 5 - 5 = -16 - 5

This simplifies to:

7m = -21

By combining like terms, we've simplified the equation and brought it closer to a form where we can easily isolate 'm'. This systematic approach is key to solving more complex equations as well.

Step 3: Isolate the Variable

Now that we have 7m = -21, the final step is to isolate 'm' completely. To do this, we need to get rid of the coefficient (the number multiplying 'm'), which is 7 in this case. We can isolate 'm' by dividing both sides of the equation by 7:

(7m) / 7 = (-21) / 7

This simplifies to:

m = -3

And there you have it! We've successfully isolated 'm' and found its value. This step is the culmination of all the previous steps and provides the solution to the equation. Dividing by the coefficient is a standard technique in solving linear equations, and it's essential for arriving at the final answer.

The Final Solution

After following these steps, we have determined that the solution to the equation 5(m+1) = -2(m+8) is:

m = -3

This means that if we substitute -3 for 'm' in the original equation, both sides of the equation will be equal. To verify this, we can substitute -3 back into the original equation and check if it holds true.

Verifying the Solution

To ensure our solution is correct, it's always a good practice to verify it by substituting the value we found for 'm' back into the original equation:

Original equation: 5(m+1) = -2(m+8)

Substitute m = -3:

5((-3)+1) = -2((-3)+8)

Simplify the left side:

5(-2) = -10

Simplify the right side:

-2(5) = -10

Since both sides of the equation are equal (-10 = -10), our solution m = -3 is correct. Verification is a critical step in problem-solving, as it helps catch any potential errors and provides confidence in your answer. By taking the time to check your work, you can be sure you've solved the equation accurately.

Common Mistakes to Avoid

When solving algebraic equations, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One frequent mistake is an error in distribution. For example, when dealing with an equation like 5(m+1) = -2(m+8), it's crucial to multiply the number outside the parentheses by each term inside. Forgetting to distribute to all terms can lead to an incorrect equation and a wrong answer.

Another common mistake is failing to combine like terms correctly. This often happens when terms are moved from one side of the equation to the other. Remember to perform the same operation on both sides to maintain the equation's balance. For instance, if you have 7m + 5 = -16, you need to subtract 5 from both sides to isolate the term with 'm'. Neglecting to do this can result in an unbalanced equation and an incorrect solution.

Sign errors are also a frequent source of mistakes. Pay close attention to the signs of numbers, especially when dealing with negative numbers. For example, in the equation -2(m+8), the negative sign must be distributed correctly. Careless handling of signs can easily lead to an incorrect solution. Finally, always double-check your work, especially after performing multiple steps. Verification, as we discussed earlier, is an invaluable tool for catching errors and ensuring accuracy in your solutions.

Practice Problems

To solidify your understanding of solving for 'm' in linear equations, let’s try a few practice problems. Working through these examples will help you apply the steps we’ve discussed and build your confidence in solving algebraic equations. Remember to follow each step systematically: distribute, combine like terms, and isolate the variable.

1. Solve for m: 3(m - 2) = 6
2. Solve for m: -4(m + 1) = 8
3. Solve for m: 2m + 5 = -3m - 10

Take your time to work through these problems, and don’t hesitate to refer back to the steps we've covered in this guide. The more you practice, the more comfortable and proficient you’ll become in solving algebraic equations. These problems will test your ability to apply the distributive property, combine like terms, and isolate the variable, all essential skills in algebra. So, grab a pencil and paper, and let's put your skills to the test!

Conclusion

In conclusion, solving for 'm' in the equation 5(m+1) = -2(m+8) involves a systematic approach of distributing, combining like terms, and isolating the variable. By following these steps carefully and avoiding common mistakes, you can confidently solve linear equations. Remember to always verify your solution to ensure accuracy. With practice, these skills will become second nature, enabling you to tackle more complex algebraic problems. Keep practicing, and you'll master the art of solving for variables in no time!

For further learning and practice, you might find helpful resources on websites like Khan Academy Algebra.