Solving Linear Equations: A Step-by-Step Guide

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Solving Linear Equations: A Step-by-Step Guide

Hey guys! Ever get stumped by a linear equation? Don't worry, it happens to the best of us. Today, we're going to break down how to solve the equation (−6m−3)=6(5m−1)(-6m - 3) = 6(5m - 1). We will go through each step to ensure you understand the process. Trust me, once you get the hang of it, it's like riding a bike!

Understanding the Equation

Let's first understand what we are dealing with. We have the equation (−6m−3)=6(5m−1)(-6m - 3) = 6(5m - 1). This is a linear equation because the highest power of the variable 'm' is 1. Our goal is to isolate 'm' on one side of the equation to find its value. Isolating the variable means getting 'm' by itself on one side, so we know exactly what it equals. This involves using algebraic manipulations like distributing, combining like terms, and performing the same operations on both sides of the equation to maintain balance.

Step 1: Distribute

The first thing we need to do is distribute the 6 on the right side of the equation. This means multiplying 6 by both terms inside the parentheses:

6∗(5m)=30m6 * (5m) = 30m and 6∗(−1)=−66 * (-1) = -6.

So, the equation becomes:

−6m−3=30m−6-6m - 3 = 30m - 6

Distribution is a fundamental concept in algebra, ensuring that each term inside the parentheses is correctly multiplied by the term outside. This step is crucial because it simplifies the equation, allowing us to combine like terms and eventually isolate the variable. Think of it like this: you're sharing the 6 with both the 5m and the -1, making sure everyone gets their fair share. Getting this step right is super important for solving the rest of the equation correctly, so double-check your multiplication!

Step 2: Combine Like Terms

Now, we want to get all the 'm' terms on one side of the equation and the constants on the other side. Let's move the '-6m' term from the left side to the right side. To do this, we add '6m' to both sides of the equation:

−6m−3+6m=30m−6+6m-6m - 3 + 6m = 30m - 6 + 6m

This simplifies to:

−3=36m−6-3 = 36m - 6

Next, let's move the '-6' from the right side to the left side. We do this by adding '6' to both sides:

−3+6=36m−6+6-3 + 6 = 36m - 6 + 6

This simplifies to:

3=36m3 = 36m

Combining like terms is like sorting your socks – you want to group the similar items together to make things easier to manage. In this case, we're grouping the 'm' terms on one side and the constant terms on the other. By adding or subtracting terms from both sides, we maintain the equation's balance while simplifying it. Remember, whatever you do to one side, you must do to the other to keep the equation fair and square. This step sets us up perfectly for isolating 'm' in the next step!

Step 3: Isolate the Variable

Now, we want to isolate 'm'. We have:

3=36m3 = 36m

To get 'm' by itself, we need to divide both sides of the equation by 36:

336=36m36\frac{3}{36} = \frac{36m}{36}

This simplifies to:

112=m\frac{1}{12} = m

So, m=112m = \frac{1}{12}

Isolating the variable is the final showdown in solving for 'm'. It's all about getting 'm' completely alone on one side of the equation. We achieve this by performing the inverse operation of whatever is being done to 'm'. In our case, 'm' is being multiplied by 36, so we divide both sides by 36. This leaves 'm' all by itself, revealing its value. This step is the culmination of all our previous efforts, bringing us to the solution we've been working towards. Make sure to double-check your division to ensure you've accurately found the value of 'm'!

Verification

To make sure our solution is correct, we can plug m=112m = \frac{1}{12} back into the original equation:

−6(112)−3=6(5(112)−1)-6(\frac{1}{12}) - 3 = 6(5(\frac{1}{12}) - 1)

Simplifying the left side:

−612−3=−12−3=−12−62=−72-\frac{6}{12} - 3 = -\frac{1}{2} - 3 = -\frac{1}{2} - \frac{6}{2} = -\frac{7}{2}

Simplifying the right side:

6(512−1)=6(512−1212)=6(−712)=−4212=−726(\frac{5}{12} - 1) = 6(\frac{5}{12} - \frac{12}{12}) = 6(-\frac{7}{12}) = -\frac{42}{12} = -\frac{7}{2}

Since both sides are equal, our solution m=112m = \frac{1}{12} is correct!

Verification is your chance to be a detective and double-check your work. By plugging the value you found for 'm' back into the original equation, you can confirm whether your solution is correct. If both sides of the equation balance out, you've nailed it! If not, it's time to revisit your steps and look for any errors. This process not only ensures accuracy but also reinforces your understanding of the equation and the solution process. Always verify to be 100% confident in your answer!

Conclusion

So, the solution to the equation (−6m−3)=6(5m−1)(-6m - 3) = 6(5m - 1) is m=112m = \frac{1}{12}. Remember to distribute, combine like terms, and isolate the variable. And always verify your answer! You've got this! This systematic approach demystifies linear equations, making them less intimidating and more manageable. With practice, you'll become more confident in your ability to solve them accurately and efficiently. Keep practicing, and soon you'll be solving linear equations like a pro!

Practice Problems

To solidify your understanding, try solving these practice problems:

  1. 2(x+3)=4x−22(x + 3) = 4x - 2
  2. −3(y−1)=6−2y-3(y - 1) = 6 - 2y
  3. 5z+4=2(z−1)5z + 4 = 2(z - 1)

Solving these equations will give you confidence in the method and help you remember the steps. Good luck, and keep practicing!

Answer to the initial question

The correct answer was not among the options A, B, C, and D.