Error Spotting: Solving For X In 5-3(x+1)=8

Let's dive into this math problem and see where Ellie might have taken a wrong turn. Our goal is to solve the equation $5-3(x+1)=8$ and prove that $x=-2$. We'll examine each step carefully to pinpoint any mistakes.

Analyzing Ellie's Steps

Here's how Ellie started:

Step 1: The Given Equation

The first step is simply stating the given equation: $5-3(x+1)=8$. This is correct; it's just the starting point of our problem. No errors here!

Step 2: Simplification - Spotting the Mistake

Now, let's look at the second step: $2(x+1)=8$. Ellie claims she simplified the original equation to get this. But how did she do it? This is where the error lies. To go from $5-3(x+1)=8$ to $2(x+1)=8$, Ellie would have to subtract 3 from 5 but failed to consider that the term $-3$ is being multiplied by the entire expression $(x+1)$.

The correct way to simplify the equation involves dealing with the term $-3(x+1)$ properly. The $5$ is a separate term and can only be combined with other constant terms after the distribution.

To correctly solve this, one must first distribute $-3$ into the parenthesis.

Why It's Wrong:

• Ellie incorrectly combined the $5$ and $-3$ before dealing with the term $-3(x+1)$.
• The order of operations (PEMDAS/BODMAS) dictates that multiplication should be done before addition or subtraction.

Correcting the Solution

Let's solve the equation correctly to see how it should be done:

1. Given: $5-3(x+1)=8$
2. Distribute: $5 - 3x - 3 = 8$
3. Combine Like Terms: $2 - 3x = 8$
4. Subtract 2 from both sides: $-3x = 6$
5. Divide by -3: $x = -2$

So, the correct solution is indeed $x = -2$, but Ellie's steps to get there were flawed.

Detailed Explanation of the Correct Steps

To ensure clarity, let's break down each correct step with detailed explanations. Starting with the given equation, our aim is to isolate $x$ on one side of the equation. This involves careful application of algebraic principles and adherence to the order of operations.

Step 1: Given

The initial equation is $5 - 3(x + 1) = 8$. This is our starting point, and no action is required other than acknowledging the problem we need to solve. It's crucial to copy the equation correctly to avoid any initial errors.

Step 2: Distribute

The next step involves distributing the $-3$ across the terms inside the parenthesis. This means multiplying $-3$ by both $x$ and $1$. Thus, $-3(x + 1)$ becomes $-3x - 3$. The equation now looks like this: $5 - 3x - 3 = 8$.

Why Distribution is Important:

Distribution is a fundamental algebraic operation that removes parentheses and allows us to combine like terms. Forgetting to distribute properly can lead to incorrect solutions, as seen in Ellie's attempt. Paying close attention to signs (positive or negative) is also critical during distribution.

Step 3: Combine Like Terms

Now, we combine the like terms on the left side of the equation. We have two constant terms: $5$ and $-3$. Combining them gives us $5 - 3 = 2$. So, the equation simplifies to $2 - 3x = 8$.

Understanding Like Terms:

Like terms are terms that have the same variable raised to the same power. In this case, $5$ and $-3$ are both constant terms and can be combined. Combining like terms simplifies the equation and brings us closer to isolating the variable $x$.

Step 4: Subtract 2 from Both Sides

To further isolate the term with $x$, we subtract $2$ from both sides of the equation. This maintains the balance of the equation and moves us closer to solving for $x$. Subtracting $2$ from both sides of $2 - 3x = 8$ gives us $-3x = 6$.

Maintaining Balance in the Equation:

It's crucial to perform the same operation on both sides of the equation to maintain equality. This principle ensures that the solution remains valid. Subtracting $2$ from both sides isolates the term with $x$ on one side and a constant on the other.

Step 5: Divide by -3

Finally, to solve for $x$, we divide both sides of the equation by $-3$. This isolates $x$ on one side and gives us the value of $x$. Dividing both sides of $-3x = 6$ by $-3$ yields $x = -2$.

The Final Solution:

Therefore, the solution to the equation $5 - 3(x + 1) = 8$ is $x = -2$. This confirms the statement we were asked to prove.

Conclusion

Ellie's first mistake occurred in Step 2 where she incorrectly simplified the equation. She subtracted 3 from 5 before distributing the -3 across the $(x+1)$ term. The correct approach involves distributing first, then combining like terms, and finally isolating $x$. By following the correct steps, we arrive at the solution $x = -2$.

For more resources on algebraic equations and solving techniques, check out Khan Academy's Algebra Section.