Solving 5(x-4) = 3x + 4 A Step-by-Step Guide
In this article, we will delve into the step-by-step solution of the equation 5(x-4) = 3x + 4. This is a fundamental algebraic problem that requires a clear understanding of the distributive property and how to isolate the variable x. By meticulously working through each step, we will arrive at the correct answer and explain the underlying principles that make this solution valid. Whether you're a student grappling with algebra or simply someone looking to refresh your math skills, this guide will provide a comprehensive understanding of how to solve this type of equation. Let's begin by outlining the initial problem and then proceeding with the solution.
Understanding the Equation
The equation we are tasked with solving is 5(x-4) = 3x + 4. This is a linear equation, meaning the highest power of the variable x is 1. To solve for x, we need to isolate it on one side of the equation. This involves using algebraic manipulations, such as distributing, combining like terms, and performing the same operations on both sides of the equation to maintain equality. The key to solving any algebraic equation is to follow the order of operations (PEMDAS/BODMAS) in reverse when isolating the variable. This ensures that we are undoing the operations in the correct order to arrive at the solution.
Before we dive into the solution, let's break down the equation into its components:
- 5(x-4): This part of the equation requires us to use the distributive property. The number 5 needs to be multiplied by both terms inside the parentheses, x and -4.
- 3x + 4: This is a linear expression where 3 is the coefficient of x and 4 is a constant term.
- =: The equals sign signifies that the expressions on both sides of the equation are equal. Our goal is to find the value of x that makes this statement true.
With a clear understanding of the equation's structure, we can now proceed to solve it step-by-step.
Step-by-Step Solution
Our primary goal in solving the equation 5(x-4) = 3x + 4 is to isolate the variable x on one side of the equation. This involves a series of algebraic manipulations that maintain the equality of both sides. Let's break down the solution into manageable steps:
Step 1: Distribute the 5
The first step is to apply the distributive property to the left side of the equation. This means multiplying the 5 by each term inside the parentheses:
5 * x = 5x
5 * -4 = -20
So, the left side of the equation becomes 5x - 20. The equation now looks like this:
5x - 20 = 3x + 4
Step 2: Move the x terms to one side
Next, we want to group the terms containing x on one side of the equation. To do this, we can subtract 3x from both sides. This keeps the equation balanced:
5x - 20 - 3x = 3x + 4 - 3x
This simplifies to:
2x - 20 = 4
Step 3: Move the constant terms to the other side
Now, we need to move the constant terms (the numbers without x) to the other side of the equation. We can do this by adding 20 to both sides:
2x - 20 + 20 = 4 + 20
This simplifies to:
2x = 24
Step 4: Isolate x
Finally, to isolate x, we need to get rid of the coefficient 2. We can do this by dividing both sides of the equation by 2:
2x / 2 = 24 / 2
This simplifies to:
x = 12
Therefore, the solution to the equation 5(x-4) = 3x + 4 is x = 12.
Verifying the Solution
It's always a good practice to verify the solution by substituting the value of x back into the original equation. This ensures that our solution is correct and that we haven't made any mistakes during the algebraic manipulations. Let's substitute x = 12 into the original equation:
5(x-4) = 3x + 4
Substitute x with 12:
5(12-4) = 3(12) + 4
Now, let's simplify both sides of the equation:
Left Side:
5(12-4) = 5(8) = 40
Right Side:
3(12) + 4 = 36 + 4 = 40
Both sides of the equation simplify to 40, which means our solution x = 12 is correct. This verification step provides confidence that we have accurately solved the equation.
Common Mistakes to Avoid
When solving equations like 5(x-4) = 3x + 4, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. Let's discuss some of the most frequent errors:
Failure to Distribute Correctly
One of the most common mistakes is not distributing the number outside the parentheses to all terms inside. In our equation, 5(x-4), the 5 must be multiplied by both x and -4. A mistake would be to multiply 5 only by x, resulting in 5x - 4 instead of the correct 5x - 20. Always double-check that you have distributed correctly to prevent this error.
Incorrectly Combining Like Terms
Another common mistake occurs when combining like terms. For example, when simplifying 5x - 20 = 3x + 4, some might incorrectly combine 5x and -20. Remember, you can only combine terms that have the same variable raised to the same power (or are constants). In this case, 5x and 3x are like terms, and -20 and 4 are like terms. Make sure to only combine terms that are alike to avoid errors.
Sign Errors
Sign errors are also a frequent source of mistakes. For instance, when moving terms across the equals sign, remember to change their signs. If you are subtracting a term on one side, you must add it to the other side, and vice versa. For example, when moving 3x from the right side to the left side in the equation 5x - 20 = 3x + 4, it becomes -3x. Pay close attention to the signs when manipulating the equation.
Dividing or Multiplying Incorrectly
The last step in solving many equations involves dividing or multiplying to isolate the variable. A common mistake here is to only divide or multiply one term on each side of the equation instead of all terms. To maintain balance, any operation must be applied to the entire side. For example, in 2x = 24, you must divide both 2x and 24 by 2 to get x = 12. Ensure that you apply the operation to all terms to avoid errors.
By being mindful of these common mistakes, you can increase your accuracy and confidence in solving algebraic equations.
Conclusion
In conclusion, we have successfully solved the equation 5(x-4) = 3x + 4 by following a systematic approach that involved distributing, combining like terms, and isolating the variable x. The correct solution, as we determined, is x = 12. We also verified this solution by substituting it back into the original equation, confirming its accuracy. Throughout the process, we highlighted the importance of understanding the fundamental principles of algebra, such as the distributive property and maintaining balance in the equation.
Furthermore, we discussed common mistakes that students and individuals often make when solving similar equations. These include errors in distribution, incorrect combination of like terms, sign mistakes, and improper division or multiplication. By being aware of these potential pitfalls, you can significantly improve your problem-solving skills and avoid making these errors.
Solving algebraic equations is a fundamental skill in mathematics and has applications in various fields, from science and engineering to economics and finance. Mastering these skills not only enhances your mathematical abilities but also improves your critical thinking and problem-solving capabilities in general. Practice is key to becoming proficient in solving equations, so continue to work on different types of problems to solidify your understanding and build confidence. With consistent effort and a clear understanding of the underlying principles, you can tackle even more complex algebraic challenges.
