Solve The Equation: 7h - 5(3h - 8) = -72

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the fascinating world of mathematics, specifically tackling an algebraic equation that might look a little intimidating at first glance. But don't sweat it, we're going to break it down step-by-step, making it super clear and easy to follow. Our mission, should you choose to accept it, is to solve the equation $7h - 5(3h - 8) = -72$. This isn't just about finding a number; it's about understanding the process, the logic, and the power of algebra to unravel unknowns. We'll explore how to simplify expressions, isolate variables, and arrive at the correct solution. So grab your calculators (or just your brains!), and let's get this math party started!

Understanding the Equation and the Goal

Alright, let's stare this equation down: $7h - 5(3h - 8) = -72$. What's our main goal here, you ask? Simple: we want to find the value of the variable 'h' that makes this entire statement true. Think of it like a balancing scale; whatever we do to one side, we must do to the other to keep it balanced. Our ultimate aim is to get 'h' all by itself on one side of the equals sign, with a number on the other. To do that, we need to systematically remove or neutralize everything else that's hanging out with 'h'. This involves a few key algebraic maneuvers that we'll get into shortly. It's crucial to remember that each step we take is a logical deduction, moving us closer to the solution without changing the fundamental truth of the equation. We're essentially peeling back the layers of complexity, one operation at a time, to reveal the hidden value of 'h'. The initial appearance of parentheses and multiple terms involving 'h' can sometimes throw people off, but by applying the standard order of operations and distribution properties, we can simplify this expression into a much more manageable form. The process itself is a great exercise in logical thinking and problem-solving, skills that are valuable far beyond the classroom.

Step 1: Distribute the -5

The first major hurdle in our equation, $7h - 5(3h - 8) = -72$, is that pesky set of parentheses. We've got a '-5' sitting right outside, just waiting to multiply everything inside. This is where the distributive property comes into play, guys. Remember, a(b + c) = ab + ac. In our case, it's -5 multiplied by (3h - 8). So, we need to multiply -5 by 3h, and then multiply -5 by -8. Let's do the math:

• -5 times 3h equals -15h.
• -5 times -8 equals +40 (remember, a negative times a negative is a positive!).

Now, let's substitute these back into our original equation. The $7h$ at the beginning stays put. So, our equation transforms from $7h - 5(3h - 8) = -72$ to $7h - 15h + 40 = -72$. See? We've already gotten rid of the parentheses, making the equation look a whole lot cleaner. This step is absolutely critical because it allows us to combine like terms in the next stage. The distribution is a fundamental algebraic concept, and mastering it is key to solving a wide range of equations. It's the gateway to simplifying complex expressions and making them easier to work with. When you distribute, always pay close attention to the signs. A common mistake is to forget that the negative sign in front of the number outside the parentheses applies to every term inside. So, multiplying -5 by -8 correctly yields a positive 40, which is a crucial detail. If you mess up this distribution, the rest of your solution will be off, no matter how well you do the subsequent steps. This initial simplification is where many problems are won or lost, so take your time and be precise here.

Step 2: Combine Like Terms

Okay, we've successfully distributed that -5. Our equation is now $7h - 15h + 40 = -72$. What's next on the agenda? It's time to simplify things further by combining what we call 'like terms'. In algebra, like terms are terms that have the same variable raised to the same power. In our equation, the terms with 'h' are $7h$ and $-15h$. These are our like terms for the variable 'h'. The '+40' is a constant term, and '-72' is also a constant term on the other side. So, let's combine the 'h' terms:

• $7h - 15h$. Think of it as having 7 apples and then taking away 15 apples. You'd end up with -8 apples, right? So, $7h - 15h = -8h.

Now, let's rewrite the equation with this combined term. We still have the '+40' on the left side and the '-72' on the right side. So, our equation becomes $-8h + 40 = -72$. Look at that! We've reduced the number of 'h' terms from two to just one. This is a huge step towards isolating 'h'. Combining like terms is like tidying up your workspace; it makes everything much more organized and easier to handle. It’s a fundamental simplification technique that applies across all areas of algebra. By grouping similar elements together, we reduce the complexity of the expression, making it simpler to manipulate. This step is particularly important when dealing with equations that have many terms, as it helps to streamline the process and prevent errors. When combining terms, remember the rules of adding and subtracting signed numbers. For $7h - 15h$, we are essentially performing the operation $7 - 15$, which results in $-8$. The variable 'h' remains attached to the result because we are combining terms that contain 'h'. The constant terms, 40 and -72, will be dealt with in the next step.

Step 3: Isolate the Variable Term

We're getting closer, people! Our equation is now $-8h + 40 = -72$. Our mission is to get that $-8h$ term all by itself on the left side. To do that, we need to get rid of the '+40'. How do we cancel out a '+40'? We do the opposite operation: subtract 40. And remember the golden rule of equations: whatever you do to one side, you must do to the other. So, we subtract 40 from both sides of the equation:

• On the left side: $-8h + 40 - 40$. The '+40' and '-40' cancel each other out, leaving us with just $-8h$.
• On the right side: $-72 - 40$. Adding two negative numbers means we move further down the number line. So, $-72 - 40 = **-112$**.

Our equation now looks like this: $-8h = -112$. We have successfully isolated the term containing 'h'. This step is all about inverse operations. To undo addition, we subtract; to undo subtraction, we add. By applying the inverse operation to both sides, we maintain the equality of the equation while moving closer to our goal. It's like carefully removing obstacles from the path to isolate the main objective. This isolation of the variable term is a critical juncture in solving any equation. It sets the stage for the final step, which is to determine the precise value of the variable itself. By consistently applying the principle of performing the same operation on both sides, we ensure that the equation remains balanced and that our subsequent calculations are valid. The key here is recognizing that '+40' is added to the '-8h' term, and thus, subtraction is the appropriate inverse operation to eliminate it from the left side. The calculation on the right side, $-72 - 40$, requires careful attention to the rules of signed numbers; subtracting a positive number is equivalent to adding a negative number, hence $-72 + (-40) = -112$. This meticulous attention to detail in each step is what leads to an accurate final solution.

Step 4: Solve for 'h'

We've reached the final boss, folks! Our equation is currently $-8h = -112$. This means '-8 multiplied by h equals -112'. To find out what 'h' is, we need to undo that multiplication by -8. What's the opposite of multiplying by -8? That's right, dividing by -8. Again, we must do this to both sides of the equation to keep it balanced:

• On the left side: $-8h / -8$. The '-8' in the numerator and the denominator cancel each other out, leaving us with just $h$.
• On the right side: $-112 / -8$. A negative number divided by a negative number results in a positive number. So, $112 ext{ divided by } 8$. Let's figure that out: $8 imes 10 = 80$. $112 - 80 = 32$. And $8 imes 4 = 32$. So, $10 + 4 = 14$. Therefore, $-112 / -8 = **14$**.

And there we have it! The solution to our equation is $h = 14$. We've successfully isolated 'h' and found its value. This final step involves the inverse operation of multiplication, which is division. By dividing both sides by the coefficient of 'h' (which is -8), we isolate 'h' and determine its numerical value. It's crucial to remember the rules for dividing signed numbers: a negative divided by a negative yields a positive. This is why -112 divided by -8 results in a positive 14. This final calculation is the culmination of all the previous simplification steps. Each prior maneuver was essential to arrive at this point where a simple division reveals the answer. The process demonstrates how algebraic manipulation, when applied correctly, can systematically reduce a complex problem to its simplest solution. The ability to perform this final division accurately, considering the signs, is the last check in ensuring the correctness of the entire process. It's a satisfying moment when the variable finally stands alone, revealing the specific value that satisfies the original equation.

Step 5: Check Your Answer (Optional but Recommended!)

So, we found that $h = 14$. But how do we know for sure it's correct? We can perform a quick check by plugging this value back into the original equation: $7h - 5(3h - 8) = -72$. Let's substitute 14 for every 'h':

• $7(14) - 5(3(14) - 8) = -72$
• First, calculate inside the parentheses: $3(14) = 42$. So, $7(14) - 5(42 - 8) = -72$.
• Now, $42 - 8 = 34$. So, $7(14) - 5(34) = -72$.
• Next, perform the multiplications: $7(14) = 98$ and $5(34) = 170$. So, $98 - 170 = -72$.
• Finally, perform the subtraction: $98 - 170$. This equals -72.

And look at that! The left side of the equation ($ -72 $) matches the right side of the equation ($ -72 $). This means our solution, $h = 14$, is absolutely correct! Checking your answer is like double-checking your work before submitting a big project. It ensures accuracy and builds confidence in your mathematical abilities. It reinforces the understanding that the variable's value is the unique number that makes the equation a true statement. This verification step is invaluable for catching any arithmetic errors or missteps in the algebraic manipulation. It transforms the solving process from a one-way street to a cyclical one where the solution is tested against the original problem, confirming its validity. The process of substituting the found value back into the initial equation and evaluating both sides requires careful application of the order of operations (PEMDAS/BODMAS), providing an additional layer of practice and reinforcing these essential mathematical rules. If the left side does not equal the right side, it's a clear signal to go back and review each step of the solution process to identify where the error occurred. This iterative approach to problem-solving is a hallmark of strong mathematical thinking.

Conclusion: You Nailed It!

So there you have it, guys! We took a seemingly complex equation, $7h - 5(3h - 8) = -72$, and with a series of logical steps – distribution, combining like terms, isolating the variable term, and finally solving for 'h' – we arrived at the solution $h = 14$. We even did a victory lap by checking our answer! This journey through solving this equation highlights the power and elegance of algebra. It's all about breaking down problems into manageable parts and using the rules of mathematics to find the truth. Remember these steps the next time you encounter an equation like this. Practice makes perfect, so keep solving, keep exploring, and keep that mathematical curiosity alive! We hope this breakdown was super helpful for you all. Catch you in the next one, Plastik Magazine readers!