Solving Algebraic Equations Find The Missing Number When B=5

Introduction

In the realm of algebra, solving equations is a fundamental skill. These algebraic equations often involve finding unknown values that, when substituted, make the equation true. This article focuses on a specific type of algebraic problem: filling in a missing number to balance an equation. Our example equation is 6b + □ = 3b + 25, where we need to find the value that should replace the square (□) when b = 5. This exercise not only reinforces basic algebraic principles but also highlights the importance of understanding variable substitution and equation manipulation. Let's delve into the step-by-step process of solving this equation, ensuring that you grasp each concept thoroughly.

Understanding the Basics of Algebraic Equations

Before we tackle the problem at hand, let's establish a firm understanding of what algebraic equations entail. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions are connected by an equals sign (=). The expressions often contain variables, which are symbols (usually letters) that represent unknown quantities. Solving an equation means finding the value(s) of the variable(s) that make the equation true. In our case, the variable is b, and we are given that b = 5. This piece of information is crucial because it allows us to substitute the value of b into the equation, simplifying it and bringing us closer to finding the missing number. The power of algebra lies in its ability to represent real-world situations mathematically, making it an indispensable tool in various fields, from science and engineering to economics and computer science. To successfully navigate algebraic equations, it's essential to be comfortable with concepts such as variables, constants, coefficients, and the order of operations. These building blocks form the foundation upon which more complex algebraic problems are solved.

Step-by-Step Solution: Finding the Missing Number

Now, let's break down the solution to the equation 6b + □ = 3b + 25 when b = 5. This step-by-step approach will clarify the process and ensure that you can apply similar methods to other algebraic problems.

Step 1: Substitute the Value of b

The first step is to substitute the given value of b, which is 5, into the equation. This means replacing every instance of b in the equation with the number 5. Our equation then becomes: 6(5) + □ = 3(5) + 25. By performing this substitution, we transform the equation from one involving a variable to one involving numerical values, making it easier to solve. Substitution is a fundamental technique in algebra, allowing us to simplify equations and isolate the unknown value we are trying to find. It is essential to carry out this step accurately, as any errors here will propagate through the rest of the solution.

Step 2: Perform the Multiplication

Next, we perform the multiplication operations on both sides of the equation. This involves multiplying 6 by 5 on the left side and 3 by 5 on the right side. The equation now looks like this: 30 + □ = 15 + 25. By completing the multiplication, we further simplify the equation, reducing the number of terms and making it more manageable. Remember to follow the order of operations (PEMDAS/BODMAS), which dictates that multiplication should be performed before addition or subtraction. This step is crucial for maintaining the balance of the equation and ensuring that we arrive at the correct solution.

Step 3: Simplify the Right Side of the Equation

Now, let's simplify the right side of the equation by adding 15 and 25. This gives us: 30 + □ = 40. Simplifying each side of the equation makes it clearer and easier to see the relationship between the terms. By combining like terms, we reduce the complexity of the equation and move closer to isolating the missing number. This step demonstrates the importance of performing arithmetic operations accurately and efficiently to arrive at the correct solution.

Step 4: Isolate the Missing Number

To find the missing number, we need to isolate it on one side of the equation. In this case, the missing number is represented by the square (□). To isolate it, we need to subtract 30 from both sides of the equation. This is because the equation currently states "30 + □ = 40", and to get the □ by itself, we need to undo the addition of 30. Subtracting 30 from both sides maintains the balance of the equation, a critical principle in algebra. The equation then becomes: □ = 40 - 30. The principle of maintaining balance is fundamental to solving algebraic equations. Any operation performed on one side of the equation must also be performed on the other side to ensure that the equality remains true.

Step 5: Calculate the Missing Number

Finally, we perform the subtraction: 40 - 30 = 10. This gives us the missing number: □ = 10. Therefore, the number that should fill the square to make the equation correct when b = 5 is 10. This final step concludes our solution process, providing the answer that satisfies the original equation. By systematically working through each step, we have successfully found the missing number and reinforced our understanding of algebraic problem-solving.

Verification: Ensuring the Solution is Correct

To ensure that our solution is correct, it's crucial to verify it by substituting the missing number back into the original equation. This step acts as a check, confirming that the value we found truly balances the equation when b = 5. Our original equation is 6b + □ = 3b + 25. We found that the missing number is 10. So, let's substitute b = 5 and the missing number 10 into the equation:

6(5) + 10 = 3(5) + 25

30 + 10 = 15 + 25

40 = 40

The equation holds true! This verification step confirms that our solution, 10, is indeed the correct missing number. Verification is an essential part of problem-solving in mathematics. It provides confidence in the accuracy of the solution and helps identify any errors that may have occurred during the process. By taking the time to verify our answer, we ensure that we have a solid understanding of the problem and its solution.

Conclusion

In conclusion, we have successfully found the missing number in the equation 6b + □ = 3b + 25 when b = 5. By following a step-by-step approach, we first substituted the value of b, then simplified the equation through multiplication and addition, and finally isolated the missing number. Our solution was verified to ensure its accuracy. This exercise highlights the importance of understanding and applying basic algebraic principles. Algebraic equations are fundamental tools in mathematics and various other disciplines. Mastering the techniques to solve them is essential for success in these fields. The ability to manipulate equations, substitute values, and maintain balance are key skills that extend beyond the classroom and into real-world problem-solving scenarios. By practicing these skills, you will build a strong foundation in algebra and be well-prepared for more advanced mathematical concepts.

This problem demonstrates the power of algebra in representing and solving real-world problems. With practice and a solid understanding of the fundamental principles, you can confidently tackle a wide range of algebraic challenges.