Solving For X: 2(6x + 4) - 6 + 2x = 3(4x + 3) + 1

by Alex Johnson 50 views

Let's dive into solving this equation step-by-step! Linear equations might seem intimidating at first, but with a clear and methodical approach, they become quite manageable. In this article, we will walk through each step of solving the equation 2(6x + 4) - 6 + 2x = 3(4x + 3) + 1, providing explanations and insights to ensure you understand the underlying principles. Our goal is not just to find the solution for x, but also to equip you with the skills and confidence to tackle similar problems on your own.

Understanding Linear Equations

Before we jump into the solution, let's briefly touch on what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called “linear” because when plotted on a graph, they form a straight line. The equation we're dealing with, 2(6x + 4) - 6 + 2x = 3(4x + 3) + 1, fits this description, making it a linear equation in one variable, x. Solving such equations involves isolating the variable on one side of the equation to determine its value.

Step-by-Step Solution

Now, let's break down the solution process into manageable steps:

1. Distribute

The first step in solving this equation is to eliminate the parentheses by distributing the constants outside them. This involves multiplying the constants by each term inside the parentheses. Applying this to our equation, we get:

  • 2 * (6x + 4) becomes 12x + 8
  • 3 * (4x + 3) becomes 12x + 9

So, our equation now looks like this: 12x + 8 - 6 + 2x = 12x + 9 + 1.

Remember, the distributive property is a fundamental concept in algebra. It states that a(b + c) = ab + ac. Mastering this property is crucial for simplifying and solving algebraic equations.

2. Combine Like Terms

Next, we need to simplify both sides of the equation by combining like terms. Like terms are terms that have the same variable raised to the same power (in this case, just x) or are constants. On the left side of the equation, we have 12x and 2x, which are like terms, and 8 and -6, which are also like terms. On the right side, 9 and 1 are like terms. Combining these, we have:

  • 12x + 2x = 14x
  • 8 - 6 = 2
  • 9 + 1 = 10

Our equation is now simplified to: 14x + 2 = 12x + 10.

Combining like terms helps to streamline the equation, making it easier to work with. This step reduces the complexity and brings us closer to isolating the variable.

3. Isolate the Variable Term

To isolate the variable term, we need to get all the terms containing x on one side of the equation and all the constants on the other side. A common approach is to subtract the smaller variable term from both sides. In our equation, we have 14x on the left and 12x on the right. Subtracting 12x from both sides will keep the coefficient of x positive:

  • 14x + 2 - 12x = 12x + 10 - 12x
  • This simplifies to 2x + 2 = 10

Now, we have the variable term 2x on the left side. Next, we need to move the constant term from the left side to the right side. To do this, we subtract 2 from both sides:

  • 2x + 2 - 2 = 10 - 2
  • This simplifies to 2x = 8

Isolating the variable term is a critical step. By strategically adding or subtracting terms from both sides, we maintain the equation's balance while moving closer to isolating x.

4. Solve for x

The final step is to solve for x by dividing both sides of the equation by the coefficient of x. In our case, the coefficient of x is 2. So, we divide both sides by 2:

  • 2x / 2 = 8 / 2
  • This simplifies to x = 4

Therefore, the solution to the equation is x = 4.

Dividing by the coefficient of x is the final step in isolating the variable and finding its value. This step unveils the solution to the equation.

Verification

It's always a good practice to verify your solution by substituting it back into the original equation. This ensures that your solution is correct and that you haven't made any errors along the way. Let's substitute x = 4 into the original equation:

Original equation: 2(6x + 4) - 6 + 2x = 3(4x + 3) + 1

Substituting x = 4:

  • 2(6 * 4 + 4) - 6 + 2 * 4 = 3(4 * 4 + 3) + 1
  • 2(24 + 4) - 6 + 8 = 3(16 + 3) + 1
  • 2(28) - 6 + 8 = 3(19) + 1
  • 56 - 6 + 8 = 57 + 1
  • 58 = 58

Since both sides of the equation are equal, our solution x = 4 is correct.

Common Mistakes to Avoid

Solving linear equations can be tricky, and there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

  1. Incorrect Distribution: A common mistake is not distributing the constant to all terms inside the parentheses. For example, in the expression 2(6x + 4), you must multiply both 6x and 4 by 2.
  2. Combining Unlike Terms: Only like terms can be combined. For example, you cannot combine 12x and 8 because they are not like terms.
  3. Incorrectly Applying Operations: When moving terms from one side of the equation to the other, remember to perform the opposite operation. For instance, if you have +2 on one side, you need to subtract 2 from both sides to move it.
  4. Forgetting to Verify: Always verify your solution by substituting it back into the original equation. This will catch any errors you may have made.

Tips for Success

To improve your skills in solving linear equations, consider the following tips:

  • Practice Regularly: The more you practice, the more comfortable you'll become with the process.
  • Show Your Work: Write down each step clearly. This makes it easier to identify and correct any mistakes.
  • Double-Check Your Work: Before moving on to the next step, double-check your calculations to ensure accuracy.
  • Understand the Concepts: Don't just memorize the steps; understand why each step is necessary.
  • Seek Help When Needed: If you're struggling, don't hesitate to ask for help from a teacher, tutor, or classmate.

Conclusion

In this article, we've walked through the process of solving the linear equation 2(6x + 4) - 6 + 2x = 3(4x + 3) + 1. By following a step-by-step approach, we simplified the equation, isolated the variable, and found the solution x = 4. Remember, practice is key to mastering these skills. Understanding the fundamental principles and avoiding common mistakes will make you a confident problem solver.

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