You know, sometimes math problems can feel like a bit of a puzzle, right? You're staring at an equation, and the goal is to find that elusive 'x'. It's like trying to find a hidden key. Let's take a look at one such puzzle: $$-2(5x + 4) = -5(x + 7)$$. Our mission, should we choose to accept it, is to find the value of 'x' and express it accurately to the nearest hundredth.
First off, I like to clear out those parentheses. It just makes things look a bit tidier. On the left side, we distribute the -2: $$-2 \times 5x$$ gives us $$-10x$$, and $$-2 \times 4$$ is $$-8$$. So, the left side becomes $$-10x - 8$$.
Now, for the right side. We distribute the -5: $$-5 \times x$$ is $$-5x$$, and $$-5 \times 7$$ is $$-35$$. That makes the right side $$-5x - 35$$.
So, our equation now looks like this: $$-10x - 8 = -5x - 35$$.
Next, we want to gather all the 'x' terms on one side and the constant numbers on the other. I usually prefer to move the 'x' terms to the side where they'll end up positive, but either way works. Let's add $$-5x$$ to both sides (which is the same as subtracting $$-5x$$). This gives us $$-10x + 5x - 8 = -35$$, which simplifies to $$-5x - 8 = -35$$.
Now, let's move the constant term, -8, to the right side. We do this by adding 8 to both sides: $$-5x = -35 + 8$$.
Combining those numbers on the right, we get $$-5x = -27$$.
Almost there! To isolate 'x', we divide both sides by -5: $$x = \frac{-27}{-5}$$.
And that simplifies to $$x = \frac{27}{5}$$.
Now, if we want to see this as a decimal, $$27 \div 5$$ is $$5.4$$.
The question specifically asks for the answer to the nearest hundredth. So, even though it's a nice, clean 5.4, we need to express it with two decimal places. That means we add a zero at the end: $$5.40$$.
It's a bit like when we talk about pi (π). We know it's roughly 3.14159..., but if we need it to the nearest hundredth, we look at that third decimal place. If it's 5 or greater, we round up; if it's less than 5, we keep it as is. In our case, after the 4, there are no more digits, so we just add a zero to show we've considered the hundredths place. So, $$x = 5.40$$.
See? Just a few steps, and we've solved the puzzle. It's all about breaking it down and taking it one step at a time.