Mistakes In Solving Two-Step Equations

Hey math whizzes and equation adventurers! Welcome back to Plastik Magazine, where we dive deep into the awesome world of numbers and problem-solving. Today, we've got a real head-scratcher courtesy of Brianna, who tackled a two-step equation but, well, let's just say she took a little detour on the way to the right answer. You know how it is, guys, even the sharpest minds can slip up sometimes, and that's totally okay! In fact, learning from mistakes is where the real magic happens. So, let's roll up our sleeves, put on our detective hats, and figure out exactly where Brianna went off the rails. This isn't just about finding the error; it's about understanding why it's an error and how to avoid it next time. We'll be dissecting her steps, explaining the concepts behind them, and making sure you guys are armed with the knowledge to conquer any two-step equation that comes your way. Get ready to boost your math game!

The Equation and Brianna's First Steps

Alright, let's set the scene. Brianna's challenge was to solve this juicy two-step equation: 0.1x + 3 = 1.7. Seems pretty straightforward, right? It involves a variable (x), a coefficient (0.1), a constant term (+3), and a result (1.7). The goal, as always, is to isolate 'x' – to get it all by itself on one side of the equals sign. This means we need to undo the operations that are being done to 'x'. First, 'x' is being multiplied by 0.1, and then 3 is being added to that result. To solve it, we generally work backward, performing the inverse operations. We'd typically start by dealing with the addition or subtraction, and then move on to the multiplication or division.

Brianna's work started like this:

0.1x + 3 = 1.7
(-3) + 0.1x + 7 = 1.7 + (-3)   Line 1

Immediately, my math senses started tingling. Line 1 is where things get a bit... interesting. Brianna decided to subtract 3 from the left side of the equation, which is a perfectly valid first step to start isolating the term with 'x'. The golden rule of equations, remember, is whatever you do to one side, you must do to the other to maintain balance. So, if she's subtracting 3 from the left, she also needs to subtract 3 from the right. That part is correct: 1.7 + (-3) is indeed on the right side. However, look closely at the left side: (-3) + 0.1x + 7. Wait a minute! Where did the + 7 come from? And why is there a -3 added before the 0.1x term? This isn't just a typo, guys; it looks like a mix-up in applying the inverse operation. She should have just subtracted 3 from both sides, leaving 0.1x on the left. Instead, she added -3 and then also added +7, which is essentially adding -3 + 7 = 4 to the left side. This is a major deviation from the correct procedure and is the root of her subsequent troubles. So, the first key takeaway here is precision when applying inverse operations. Make sure you're only performing the necessary step to undo the operation affecting the variable term.

Decoding Line 2: The Decimal Debacle

Now, let's examine Line 2, where Brianna simplifies her equation after Line 1:

0.1x = -1.3   Line 2

This line is supposed to be the result of simplifying Line 1. Let's see what should have happened if Brianna had followed the correct first step (subtracting 3 from both sides of 0.1x + 3 = 1.7). The correct first step would look like this:

0.1x + 3 - 3 = 1.7 - 3
0.1x = -1.3

Hey, look at that! The result 0.1x = -1.3 is correct, but only if the preceding step (Line 1) was performed correctly. The problem is, Brianna's Line 1 was not correct. She introduced a +7 and a -3 on the left side, which, when simplified, gives 0.1x + 4. So, if she had correctly simplified her own incorrect Line 1, she should have had 0.1x + 4 = -1.3. Then, subtracting 4 from both sides would yield 0.1x = -5.3. So, while the value of Line 2 (0.1x = -1.3) matches what we'd get from the correct first step, it's a coincidence stemming from a flawed initial operation. This highlights how crucial each step is. Sometimes, you might stumble upon the right intermediate result through a series of errors, but it's not a reliable strategy. The key here is to always check your work and ensure that each simplification logically follows from the previous step and correctly applies the principles of inverse operations. Brianna got lucky here, but this kind of accidental correctness can mask deeper misunderstandings. Understanding why 0.1x = -1.3 is the correct simplification of 0.1x + 3 = 1.7 is fundamental. It's about isolating the 0.1x term by eliminating the +3 using its inverse, -3, on both sides.

The Final Push: Solving for x

Now we're at the brink of solving for 'x'. We have the equation from Line 2: 0.1x = -1.3. This is a simple one-step equation now. The variable 'x' is being multiplied by 0.1, and to isolate 'x', we need to perform the inverse operation: division. We need to divide both sides of the equation by 0.1.

Let's see what Brianna did next:

(-0.1) + 0.1x = -1.3 + (-0.1)   Line 3

Facepalm. Okay guys, this is where the wheels really come off. Brianna is still stuck in the