Solving $7h - 5(3h - 8) = -72$ A Step-by-Step Guide

Solving algebraic equations is a fundamental skill in mathematics. It involves isolating the unknown variable, in this case, 'h', by performing operations on both sides of the equation while maintaining equality. This comprehensive guide provides a detailed, step-by-step solution to the equation $7h - 5(3h - 8) = -72$, along with explanations of the underlying principles and strategies involved. This approach ensures that you not only understand the solution to this specific problem but also gain the knowledge and skills to tackle similar equations confidently. The process involves simplifying expressions, applying the distributive property, combining like terms, and using inverse operations to isolate the variable. By mastering these techniques, you'll be well-equipped to solve a wide range of algebraic equations. Let's embark on this journey of mathematical discovery and unravel the solution together. This step-by-step guide will help you not only solve the equation but also understand the logic behind each operation. Solving equations is a cornerstone of algebra, and a solid grasp of these concepts will be invaluable in your mathematical journey. Understanding the 'why' behind each step is as important as knowing the 'how'.

1. Parentheses: Distribute

The first step in solving the equation $7h - 5(3h - 8) = -72$ involves addressing the parentheses. To do this, we apply the distributive property, which states that $a(b + c) = ab + ac$. In our case, we need to distribute the -5 across the terms inside the parentheses, $(3h - 8)$. This means multiplying -5 by both $3h$ and $-8$. This process is crucial for simplifying the equation and bringing us closer to isolating the variable 'h'. The distributive property is a fundamental concept in algebra and is essential for working with expressions that contain parentheses. By correctly applying this property, we can remove the parentheses and create a more manageable equation. Remember, paying close attention to signs is vital when distributing negative numbers. A simple mistake in sign can lead to an incorrect solution. This initial distribution sets the stage for the subsequent steps in solving the equation, so accuracy and understanding are key. Carefully performing this step will make the rest of the solution process smoother and more straightforward. Distributing the -5 across the parentheses transforms the equation into a form where we can combine like terms and eventually isolate the variable.

Applying the distributive property, we get:

$7h - 5(3h - 8) = 7h - 5 * 3h - 5 * (-8)$ $7h - 15h + 40 = -72$

2. Combine Terms

After distributing, the equation becomes $7h - 15h + 40 = -72$. The next crucial step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, $7h$ and $-15h$ are like terms because they both contain the variable 'h' raised to the power of 1. To combine them, we simply add their coefficients: $7 - 15 = -8$. This step simplifies the equation further, making it easier to isolate the variable 'h'. Combining like terms is a fundamental algebraic technique that streamlines equations and helps to reveal the underlying structure. It's like organizing your tools before starting a project; it makes the whole process more efficient. By grouping similar terms together, we reduce the complexity of the equation and move closer to the solution. Make sure to pay close attention to the signs of the coefficients when combining like terms, as a sign error can significantly impact the final answer. This step is a bridge between the initial distribution and the subsequent isolation of the variable, so a firm understanding of this process is essential for solving algebraic equations effectively.

Combining the 'h' terms, we have:

$(7h - 15h) + 40 = -72$ $-8h + 40 = -72$

3. Apply Properties: Subtract

Now we have the simplified equation $-8h + 40 = -72$. Our goal is to isolate 'h', which means getting it by itself on one side of the equation. To do this, we need to undo the operations that are being performed on 'h'. Currently, 'h' is being multiplied by -8 and then 40 is being added. We'll tackle the addition first by using the inverse operation, which is subtraction. To maintain the equality of the equation, we must subtract 40 from both sides. This is a key principle in solving equations: whatever operation you perform on one side, you must perform on the other. This ensures that the equation remains balanced and the solution remains valid. Subtracting 40 from both sides will eliminate the constant term on the left side and bring us closer to isolating 'h'. Understanding the concept of inverse operations is fundamental to solving equations. Each mathematical operation has an inverse that undoes it (addition and subtraction, multiplication and division). By strategically applying these inverse operations, we can systematically isolate the variable and find its value. This process is like peeling away the layers of an onion, gradually revealing the core – the value of 'h'. The ability to choose and apply the correct inverse operation is a cornerstone of algebraic problem-solving.

Subtracting 40 from both sides:

$-8h + 40 - 40 = -72 - 40$ $-8h = -112$

4. Apply Properties: Divide

We've arrived at the equation $-8h = -112$. The variable 'h' is now being multiplied by -8. To isolate 'h', we need to undo this multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the equation by -8. Remember, maintaining the balance of the equation is crucial, so we perform the same operation on both sides. Dividing by -8 will effectively cancel out the -8 on the left side, leaving 'h' by itself. This step brings us to the final solution of the equation. The principle of using inverse operations is once again at play here, demonstrating its fundamental importance in solving algebraic equations. Understanding this principle allows you to systematically work towards isolating the variable, regardless of the complexity of the equation. Paying close attention to the signs is particularly important when dividing by a negative number. A mistake in sign can lead to an incorrect answer. This final division is the culmination of all the previous steps, and it reveals the value of the unknown variable 'h'. The solution we obtain represents the value of 'h' that makes the original equation true.

Dividing both sides by -8:

$\frac{-8h}{-8} = \frac{-112}{-8}$ $h = 14$

Solution

Therefore, the solution to the equation $7h - 5(3h - 8) = -72$ is $h = 14$. This means that if we substitute 14 for 'h' in the original equation, the equation will hold true. We can verify this by plugging 14 back into the original equation and checking if both sides are equal. This verification step is a valuable tool for ensuring the accuracy of our solution. It provides a sense of closure and confidence in our answer. The solution $h = 14$ represents a specific value that satisfies the equation, and it is the only value that does so. Solving equations is not just about finding a number; it's about finding the value that makes the equation a true statement. The process we've followed, from distributing to combining like terms to applying inverse operations, is a general strategy that can be applied to a wide range of algebraic equations. By mastering this strategy, you'll be well-equipped to tackle more complex mathematical problems in the future. Remember, practice is key to building your problem-solving skills, so continue to work through different types of equations to solidify your understanding.

To check our solution, substitute $h = 14$ into the original equation:

$7(14) - 5(3(14) - 8) = -72$ $98 - 5(42 - 8) = -72$ $98 - 5(34) = -72$ $98 - 170 = -72$ $-72 = -72$

The solution is correct.

5. Reset

To start over and solve a new equation or re-attempt this one, simply clear your work and begin again with the initial equation. Remember to follow the same steps: distribute, combine like terms, and apply inverse operations to isolate the variable. Practice makes perfect, so don't hesitate to try different equations and reinforce your understanding of the process. Each equation presents a unique challenge and opportunity for growth. The more you practice, the more confident and proficient you'll become in solving algebraic equations. If you encounter difficulties, revisit the steps outlined in this guide and carefully analyze each operation. Identifying the specific point of confusion is crucial for overcoming obstacles and developing a deeper understanding of the concepts involved. Don't be discouraged by mistakes; they are valuable learning experiences. Embrace the challenge and continue to refine your problem-solving skills. With persistence and a systematic approach, you can master the art of solving equations.