Solve The Equation `2x + 28 = 40`, Showing Each Step.

Jun 18, 2025 by ADMIN 54 views

Practice Using Equations to Solve Problems

Part F: Solving Fran's Equation for Rectangle Width

In this section, we will delve into the practical application of equations to solve real-world problems. Specifically, we will focus on solving Fran's equation, 2x + 28 = 40, to determine the width of a rectangle. This exercise underscores the fundamental algebraic principles involved in isolating variables and arriving at solutions.

Step-by-Step Solution

The equation 2x + 28 = 40 represents a scenario where x is the unknown width we aim to find. Our goal is to isolate x on one side of the equation to determine its value. We will accomplish this through a series of algebraic manipulations, ensuring that we maintain the equation's balance by performing the same operations on both sides.

Isolate the term with the variable: The initial step involves eliminating the constant term on the side of the equation with the variable. In this case, we subtract 28 from both sides:
```
2x + 28 - 28 = 40 - 28
```
This simplifies to:
```
2x = 12
```
This step is rooted in the subtraction property of equality, which states that subtracting the same number from both sides of an equation preserves the equality. By subtracting 28, we effectively isolate the term 2x on the left side, bringing us closer to solving for x.
Solve for the variable: Now that we have 2x = 12, we need to isolate x. Since x is multiplied by 2, we perform the inverse operation, which is division. We divide both sides of the equation by 2:
```
(2x) / 2 = 12 / 2
```
This simplifies to:
```
x = 6
```
This step employs the division property of equality, which allows us to divide both sides of an equation by the same non-zero number without altering the equation's balance. Dividing by 2 isolates x, giving us its value.

Interpretation of the Solution

The solution x = 6 represents the width of the rectangle in the context of Fran's problem. It is crucial to understand that this numerical result has a real-world interpretation. If the problem context specifies units (e.g., centimeters, inches), we must include those units in our answer to provide a complete and meaningful solution. For instance, if the dimensions were in centimeters, the width of the rectangle would be 6 centimeters.

Importance of Showing Work

The problem explicitly asks us to show our work. This is not merely a procedural requirement but a critical aspect of problem-solving in mathematics. Showing our work allows others (and ourselves) to follow our reasoning, identify any errors, and understand the steps taken to arrive at the solution. It also demonstrates a clear and logical thought process, which is highly valued in mathematical problem-solving.

In this case, showing the steps:

2x + 28 - 28 = 40 - 28
2x = 12
(2x) / 2 = 12 / 2
x = 6

provides a transparent and verifiable solution. Each line represents a specific algebraic manipulation, making it easy to trace the solution path and ensure its correctness.

Checking the Solution

To ensure the accuracy of our solution, it is always a good practice to check it. We can do this by substituting the value we found for x back into the original equation and verifying that it holds true. Substituting x = 6 into 2x + 28 = 40, we get:

2(6) + 28 = 40
12 + 28 = 40
40 = 40

Since the equation holds true, we can confidently conclude that our solution x = 6 is correct. This verification step adds an extra layer of assurance and reinforces the reliability of our problem-solving process.

Connecting to Real-World Applications

Solving equations like this is not just an abstract mathematical exercise; it has numerous real-world applications. Understanding how to manipulate equations to solve for unknowns is crucial in various fields, including engineering, physics, economics, and computer science. For instance, engineers might use equations to calculate the dimensions of a structure, physicists might use them to determine the trajectory of a projectile, and economists might use them to model market behavior.

In this specific example, Fran's equation relates to finding the dimensions of a rectangle. This type of problem can arise in various practical situations, such as designing a room layout, calculating the amount of material needed for a project, or optimizing the use of space. By mastering the techniques for solving equations, we equip ourselves with valuable tools for tackling a wide range of real-world challenges.

Common Pitfalls and How to Avoid Them

When solving equations, several common pitfalls can lead to errors. Being aware of these pitfalls and developing strategies to avoid them is essential for accurate problem-solving.

Incorrectly applying the order of operations: One common mistake is failing to follow the correct order of operations (PEMDAS/BODMAS). When simplifying expressions, it is crucial to perform operations in the correct sequence: parentheses/brackets, exponents/orders, multiplication and division (from left to right), and addition and subtraction (from left to right). In the equation 2x + 28 = 40, we correctly addressed the addition before dealing with the multiplication by first subtracting 28 from both sides.
Performing operations on only one side of the equation: A fundamental principle of solving equations is maintaining balance. Any operation performed on one side must also be performed on the other side. For example, if we subtract 28 from the left side, we must also subtract 28 from the right side. Failing to do so will disrupt the equation's balance and lead to an incorrect solution.
Making arithmetic errors: Simple arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can derail the solution process. It is crucial to double-check each step and use tools like calculators when necessary to minimize these errors. In our solution, we carefully performed the subtraction 40 - 28 = 12 and the division 12 / 2 = 6 to ensure accuracy.
Not simplifying properly: Before isolating the variable, it is often necessary to simplify the equation by combining like terms. Failing to simplify can make the equation more complex and increase the likelihood of errors. In our example, the equation was relatively simple, but in more complex scenarios, simplification is a critical step.
Forgetting to check the solution: As mentioned earlier, checking the solution by substituting it back into the original equation is a crucial step in verifying its accuracy. Skipping this step can lead to accepting an incorrect solution. We demonstrated the importance of checking by substituting x = 6 back into 2x + 28 = 40 and confirming that the equation held true.

By being mindful of these common pitfalls and adopting strategies to avoid them, we can significantly improve our accuracy and confidence in solving equations.

Conclusion

Solving Fran's equation, 2x + 28 = 40, to find the width of the rectangle demonstrates the practical application of algebraic principles. By systematically isolating the variable through valid algebraic manipulations and verifying the solution, we arrive at the correct answer, x = 6. This exercise not only reinforces our understanding of equation-solving techniques but also highlights their relevance in real-world contexts. The ability to solve equations is a fundamental skill in mathematics and a valuable asset in various fields and everyday problem-solving scenarios.

By understanding the step-by-step solution, interpreting the result in context, showing our work, and checking our solution, we can confidently tackle similar problems in the future. Remember, the key to success in mathematics lies in practice and a thorough understanding of the underlying concepts.

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