Alice's Equation Error: Find The Mistake!

Oct 29, 2025 by ADMIN 42 views

Hey guys! Let's break down this math problem and see where Alice might have gone wrong. Math can be tricky, and even a small mistake can throw off the whole answer. Our goal is to carefully examine each step Alice took and pinpoint any errors. So, grab your pencils, and let's get started!

Analyzing Alice's Solution

Alice was trying to solve the equation: $-18 - 3x = 12$ . Let's walk through her steps:

Step 1: $-18 - 3x = 12$
Step 2: $-3x = 6$
Step 3: $x = -2$

Now, let's figure out if Alice made any mistakes along the way.

Step-by-Step Breakdown

Let's go through each step in detail to see if we can find any errors.

Original Equation: $-18 - 3x = 12$

The original equation is our starting point. No changes here, so everything looks good so far.

Step 1 to Step 2: Isolating the Term with 'x'

The goal here is to isolate the term with 'x' (which is $-3x$ ) on one side of the equation. To do this, we need to get rid of the $-18$ on the left side. We can do this by adding $18$ to both sides of the equation. So, the correct transformation should be:

$-18 - 3x + 18 = 12 + 18$

This simplifies to:

$-3x = 30$

Okay, now let's compare this to what Alice did. Alice went from $-18 - 3x = 12$ to $-3x = 6$ . This is where the error lies! Instead of adding 18 to both sides, it seems Alice subtracted 18 from the right side or made some other mistake. The correct value on the right side should be $12 + 18 = 30$ , not $6$ .

Step 2 to Step 3: Solving for 'x'

Alice went from $-3x = 6$ to $x = -2$ . Let's analyze this step, assuming that $-3x = 6$ was correct (which it isn't, but let's pretend for a moment). To solve for 'x', we would need to divide both sides of the equation by $-3$ :

$x = \frac{6}{-3}$

$x = -2$

So, the math in this step is correct, given the incorrect value from the previous step. If the equation were indeed $-3x = 6$ , then $x = -2$ would be the correct solution. However, since Step 2 is wrong, this correct calculation is based on a faulty premise.

Identifying the Error

So, where did Alice go wrong? The error occurred when she went from the original equation $-18 - 3x = 12$ to $-3x = 6$ . The correct transformation should have been to add $18$ to both sides, resulting in $-3x = 30$ .

Therefore, the statement that is true is:

The 6 should be 30 in step two.

Why the Other Options Are Incorrect

Let's quickly look at why the other options are wrong:

The 6 should be -4 in step two: This is incorrect because there's no mathematical operation that would lead to $-4$ as the constant term when isolating $-3x$ .
Alice is correct: This is incorrect because, as we've shown, Alice made a mistake in Step 2.
The 6 should be -6 in step two: This is also incorrect. While a negative number is involved in solving for x, the correct constant term after adding 18 to both sides of the original equation is positive 30.

The Correct Solution

For completeness, let's solve the equation correctly:

$-18 - 3x = 12$

Add 18 to both sides:

$-3x = 12 + 18$

$-3x = 30$

Divide both sides by -3:

$x = \frac{30}{-3}$

$x = -10$

So, the correct solution is $x = -10$ , not $x = -2$ .

Key Takeaways

Careful Arithmetic: Always double-check your arithmetic, especially when dealing with negative numbers.
Isolate the Variable: The key to solving for a variable is to isolate it on one side of the equation.
Perform the Same Operation on Both Sides: Whatever you do to one side of the equation, you must do to the other side to maintain equality.

Mastering Linear Equations

Understanding how to solve linear equations is fundamental in algebra. These equations pop up everywhere, from simple word problems to more complex scientific calculations. The key is to follow the correct steps and be meticulous with your arithmetic. Here's a more detailed look at the process:

1. Simplify Both Sides

Before you start moving terms around, make sure each side of the equation is as simple as possible. This means combining like terms and distributing any multiplication over parentheses.

For example, if you have something like $2(x + 3) - 5 = 7x - 2x + 1$ , you'd first distribute the 2 on the left: $2x + 6 - 5 = 7x - 2x + 1$ . Then, combine like terms on both sides: $2x + 1 = 5x + 1$ .

2. Isolate the Variable Term

Get all the terms containing the variable you're solving for (in this case, 'x') on one side of the equation. You can do this by adding or subtracting terms from both sides. The goal is to have something like 'ax = b', where 'a' and 'b' are constants.

In our example, $2x + 1 = 5x + 1$ , we can subtract $2x$ from both sides to get $1 = 3x + 1$ . Then, subtract 1 from both sides to get $0 = 3x$ .

3. Solve for the Variable

Once you have the variable term isolated, divide both sides of the equation by the coefficient of the variable (the number multiplying the variable). This will give you the value of the variable.

In our example, $0 = 3x$ , we divide both sides by 3 to get $x = 0$ .

Common Mistakes to Avoid

Forgetting to Distribute: Make sure you distribute multiplication over all terms inside parentheses. For example, $3(x + 2) = 3x + 6$ , not $3x + 2$ .
Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine $3x$ and $5x$ to get $8x$ , but you can't combine $3x$ and $5x^2$ .
Incorrectly Applying Operations: Remember to perform the same operation on both sides of the equation to maintain equality. If you add 5 to one side, you must add 5 to the other side.
Sign Errors: Be careful with negative signs. Make sure you distribute them correctly and keep track of them when adding or subtracting terms.

Practice Makes Perfect

The best way to master solving linear equations is to practice. Work through a variety of problems, starting with simple ones and gradually moving to more complex ones. Pay attention to your steps and double-check your work to avoid making mistakes.

Conclusion

In conclusion, Alice made a mistake in the second step of solving the equation. By adding 18 to both sides, she should have obtained -3x = 30, not -3x = 6. Therefore, the correct answer is that the 6 should be 30 in step two. Always remember to double-check each step and pay close attention to the operations you're performing. Happy solving, folks!