Karen's Solution To X - 7 + 5x = 36 Unraveling The Steps

Understanding the Problem

The problem presents us with a linear equation: x - 7 + 5x = 36. Our goal is to isolate the variable x and find its value. Karen believes the solution is x = 6, and we need to determine which method she might have used to arrive at this answer. The provided options describe different algebraic manipulations, and we'll analyze each one to see if it leads to the correct solution.

Let's delve deeper into the process of solving equations. The fundamental principle behind solving any equation is maintaining balance. Whatever operation we perform on one side of the equation, we must perform the same operation on the other side. This ensures that the equality remains valid. In this case, we're dealing with a linear equation, which means the highest power of the variable x is 1. Solving linear equations typically involves combining like terms, isolating the variable term, and then isolating the variable itself. There are several valid approaches, and Karen may have used one of them. It's crucial to understand the order of operations and the properties of equality to effectively solve equations like this. The following sections will explore the steps involved in solving the given equation and compare them to the methods suggested in the options.

Analyzing the Options

To identify Karen's method, we'll examine each option individually and see if it logically leads to the solution x = 6. We will perform the steps outlined in each option and check if the equation transforms in a way that isolates x. It's important to remember that there might be multiple ways to solve the equation, but only one option will accurately describe a valid method. By carefully analyzing each step, we can determine which option Karen likely used.

Detailed Solution Steps

Let's solve the equation x - 7 + 5x = 36 step by step to understand the process and evaluate the given options effectively. The most straightforward approach involves combining like terms first. We can combine the x terms, which are x and 5x, resulting in 6x. This simplifies the equation to 6x - 7 = 36. Next, we want to isolate the term with x, so we need to get rid of the -7 on the left side. To do this, we add 7 to both sides of the equation. This gives us 6x - 7 + 7 = 36 + 7, which simplifies to 6x = 43. Finally, to isolate x, we divide both sides of the equation by 6, resulting in x = 43/6. This is the actual solution to the equation.

Now, let's analyze the solution x = 43/6. It's a fraction, and it's not equal to 6, which is what Karen found. This indicates that Karen made a mistake somewhere in her calculation, or the problem statement contains an error. However, our goal is to determine which method she could have used, not necessarily the method that leads to the correct solution. We will proceed by evaluating the options based on the steps they describe, even if those steps might lead to an incorrect answer in this specific case.

Option A: Add x + 5x, Subtract 7 from Both Sides

This option suggests two steps: first, combining the x terms (x + 5x), and second, subtracting 7 from both sides of the equation. Let's analyze each step.

The first step, adding x + 5x, is a valid algebraic manipulation. Combining these like terms gives us 6x. So, the equation x - 7 + 5x = 36 becomes 6x - 7 = 36. This is a correct simplification of the original equation. Now, let's consider the second step: subtracting 7 from both sides of the equation. Starting with 6x - 7 = 36, if we subtract 7 from both sides, we get 6x - 7 - 7 = 36 - 7, which simplifies to 6x - 14 = 29. This step, while mathematically valid, does not move us closer to isolating x. In fact, it complicates the equation further. To correctly isolate x, we should be adding 7 to both sides, not subtracting.

Therefore, while the first part of this option is correct, the second part introduces an error. Subtracting 7 from both sides is not the correct operation to isolate x. This suggests that Option A is not the method Karen likely used, as it deviates from the standard procedure for solving linear equations.

Option B: Add x - 7 + 5x, Add 36 to Both Sides

This option proposes adding the entire left side of the equation (x - 7 + 5x) to both sides, and then adding 36 to both sides. Let's break down the implications of these steps.

The first part, adding x - 7 + 5x to both sides, is a highly unusual and ineffective approach to solving equations. Starting with x - 7 + 5x = 36, adding the left side to both sides results in: (x - 7 + 5x) + (x - 7 + 5x) = 36 + (x - 7 + 5x). This simplifies to 2(x - 7 + 5x) = 36 + x - 7 + 5x. Further simplification gives us 2(6x - 7) = 3x + 29, which expands to 12x - 14 = 6x + 29. This operation doesn't help in isolating x; it actually makes the equation more complex.

The second part, adding 36 to both sides, is equally problematic. Starting from the already complicated equation 12x - 14 = 6x + 29, adding 36 to both sides results in 12x - 14 + 36 = 6x + 29 + 36, which simplifies to 12x + 22 = 6x + 65. This step further deviates from the goal of isolating x. Adding constants to both sides in this manner doesn't eliminate terms or simplify the equation in a useful way.

Option B describes a method that is not only incorrect but also illogical for solving linear equations. Adding the entire left side to both sides and then adding a constant is not a standard algebraic technique and does not lead to the solution. Therefore, it's highly unlikely that Karen used this method.

Option C: Add -7 and 5x

This option suggests adding -7 and 5x. This statement is a bit ambiguous and needs careful interpretation. It's likely that this option refers to the initial step of combining like terms, but it doesn't explicitly state that. Let's analyze the possibilities and see if it could lead to Karen's solution of x = 6.

The phrase "add -7 and 5x" could be interpreted as an isolated operation, meaning we are simply adding these two terms together. However, this doesn't make sense in the context of solving an equation. We need to perform operations on both sides of the equation to maintain balance. Therefore, a more plausible interpretation is that this option is referring to the process of combining the constant term (-7) with the x term (5x) on the left side of the equation. However, these are not like terms and cannot be directly combined.

Going back to the original equation, x - 7 + 5x = 36, the correct first step is to combine the x terms (x and 5x) to get 6x. The equation then becomes 6x - 7 = 36. Option C doesn't explicitly mention combining the x terms, and it incorrectly implies that -7 and 5x can be directly added. This suggests a misunderstanding of how to simplify expressions and solve equations.

However, let's consider a scenario where Karen made a mistake and thought she was combining terms correctly. If she incorrectly added -7 and 5x, she might have made an error in her subsequent steps as well. To arrive at x = 6, she would have needed to manipulate the equation in a specific way. Let's see if we can reverse-engineer a possible (though incorrect) solution path.

If we substitute x = 6 into the original equation, we get 6 - 7 + 5(6) = 36, which simplifies to 6 - 7 + 30 = 36, and further to 29 = 36. This is clearly false, indicating that x = 6 is not the correct solution. However, this doesn't rule out the possibility that Karen used a flawed method that led her to this incorrect answer. We still need to assess if Option C could be part of that flawed method. Since Option C is ambiguous and doesn't describe a valid step in solving the equation, it's less likely to be the method Karen used compared to methods that describe correct initial steps, even if they are followed by errors.

Re-evaluating the Options and Identifying Karen's Potential Method

After analyzing each option, it's clear that none of them perfectly describe a correct method for solving the equation x - 7 + 5x = 36. However, we're looking for the option that could be the way Karen arrived at the solution x = 6, even if that solution is incorrect.

Option A, which suggests adding x + 5x and then subtracting 7 from both sides, starts with a correct step (combining x terms) but then introduces an error by subtracting 7 instead of adding it. This makes it a plausible method, as a simple arithmetic mistake could lead to an incorrect solution.

Option B, which involves adding the entire left side of the equation to both sides and then adding 36, is highly illogical and doesn't resemble any standard algebraic technique. It's very unlikely that Karen would have used this method.

Option C, which ambiguously states "add -7 and 5x," is the least clear and doesn't accurately describe a valid step in solving the equation. It's difficult to construct a scenario where this instruction would lead to a reasonable (even if incorrect) solution path.

Therefore, based on our analysis, Option A is the most likely candidate for the method Karen used. While it doesn't lead to the correct solution, it involves a valid initial step (combining like terms) followed by a common mistake (subtracting instead of adding). This suggests that Karen understood the basic principles of solving equations but made an arithmetic error along the way.

Conclusion

While the correct solution to the equation x - 7 + 5x = 36 is x = 43/6, Karen found the solution to be x = 6. This indicates that she likely made a mistake in her solution process. Of the given options, Option A, "add x + 5x, subtract 7 from both sides of the equation," is the most plausible method she might have used. This option includes a correct initial step (combining like terms) but then introduces an error by subtracting 7 from both sides instead of adding it. This type of mistake is common when solving equations and could easily lead to an incorrect solution. While it's important to strive for accuracy in mathematics, this problem highlights the importance of understanding the process and identifying potential errors.