Solving 45s + 16 = 151 The First Two Steps Explained
In mathematics, solving equations is a fundamental skill. It involves isolating the variable to find its value. This often requires performing a series of operations on both sides of the equation to maintain equality while simplifying the expression. When we encounter an equation like 45s + 16 = 151, the first steps we take are crucial for arriving at the correct solution. Understanding these initial steps is essential for students and anyone looking to brush up on their algebra skills. This article will thoroughly explain the process, ensuring clarity and building a solid foundation for solving more complex equations.
Understanding the Equation 45s + 16 = 151
Before diving into the steps, let’s break down the equation 45s + 16 = 151. Here, s represents the variable we need to solve for. The equation tells us that 45 times s, plus 16, equals 151. Our goal is to isolate s on one side of the equation. To do this, we must reverse the operations applied to s, following the order of operations in reverse (PEMDAS/BODMAS). The key is to perform the same operations on both sides of the equation to maintain balance.
Step 1: Isolating the Variable Term
The first crucial step in solving the equation 45s + 16 = 151 is to isolate the term containing the variable, which in this case is 45s. To do this, we need to eliminate the constant term that is being added to it, which is +16. According to the principles of algebraic manipulation, we can achieve this by performing the inverse operation. Since 16 is being added to 45s, the inverse operation is subtraction. Therefore, we subtract 16 from both sides of the equation. This ensures that the equation remains balanced, as whatever operation is performed on one side must also be performed on the other side to maintain equality.
Subtracting 16 from both sides gives us:
45s + 16 - 16 = 151 - 16
This simplifies to:
45s = 135
By subtracting 16 from both sides, we have successfully isolated the term 45s on the left side of the equation. This is a significant step forward in solving for s because we have removed the constant term that was complicating the equation. Isolating the variable term is a common strategy in solving algebraic equations, and it sets the stage for the next step in the solution process. Understanding this step is crucial for anyone looking to master algebraic problem-solving.
Step 2: Solving for the Variable
After isolating the term 45s in the previous step, we now have the simplified equation 45s = 135. The next crucial step is to isolate the variable s itself. Currently, s is being multiplied by 45. To undo this multiplication and solve for s, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 45. This ensures that the equation remains balanced, a fundamental principle in algebraic manipulations.
Dividing both sides by 45 gives us:
(45s) / 45 = 135 / 45
This simplifies to:
s = 3
By dividing both sides of the equation by 45, we have successfully isolated s and found its value. The result s = 3 is the solution to the original equation 45s + 16 = 151. This means that if we substitute 3 for s in the original equation, the equation will hold true. Solving for the variable is the ultimate goal in solving equations, and this step demonstrates the final action needed to achieve that goal. It reinforces the importance of inverse operations and maintaining balance in algebraic manipulations. This understanding is essential for tackling more complex equations in the future.
Detailed Explanation of the Correct Steps
To clarify, let's walk through the correct steps with a detailed explanation. Our goal is to solve the equation 45s + 16 = 151 for s.
Step 1: Subtract 16 from Both Sides
The first step is to isolate the term with the variable, which is 45s. To do this, we need to remove the constant term, which is +16. The inverse operation of addition is subtraction, so we subtract 16 from both sides of the equation. This maintains the equality of the equation:
45s + 16 - 16 = 151 - 16
This simplifies to:
45s = 135
Subtracting 16 from both sides is a crucial first step because it begins the process of isolating the variable term. By performing the same operation on both sides, we ensure that the equation remains balanced. This step is a fundamental technique in solving algebraic equations.
Step 2: Divide Both Sides by 45
Now that we have isolated the term 45s, the next step is to isolate s itself. Currently, s is being multiplied by 45. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 45:
(45s) / 45 = 135 / 45
This simplifies to:
s = 3
Dividing both sides by 45 is the final step in solving for s. It isolates the variable and provides the solution to the equation. This step demonstrates the importance of using inverse operations to solve for a variable in algebraic equations. The solution s = 3 means that if we substitute 3 for s in the original equation, the equation will hold true, confirming our solution.
Common Mistakes to Avoid
When solving equations, it’s easy to make mistakes if you’re not careful. One common error is not performing the same operation on both sides of the equation, which can lead to an unbalanced and incorrect solution. Another mistake is performing operations in the wrong order. Remember to follow the reverse order of operations (PEMDAS/BODMAS) when isolating the variable. For instance, it's essential to address addition and subtraction before multiplication and division.
In the context of the equation 45s + 16 = 151, a frequent mistake is dividing by 45 before subtracting 16. This violates the order of operations and complicates the equation unnecessarily. Another common error is incorrectly applying the arithmetic, such as adding 16 instead of subtracting or miscalculating the division. Attention to detail and a systematic approach can help avoid these pitfalls. It’s always a good practice to double-check your work by substituting the solution back into the original equation to ensure it holds true. This verification step can catch many simple errors and reinforce the correctness of your solution.
Why These Steps Are Important
The initial steps of subtracting 16 and then dividing by 45 are critical in solving the equation 45s + 16 = 151 because they systematically isolate the variable s. Subtracting 16 first removes the constant term from the left side of the equation, bringing us closer to isolating the term with s. This step is essential because it simplifies the equation, making it easier to manage. Dividing by 45 then isolates s completely, giving us its value. Without these steps, we wouldn't be able to determine the value of s that satisfies the equation.
These steps also demonstrate the fundamental principles of algebraic manipulation: performing inverse operations and maintaining balance. Each operation performed on one side of the equation must be mirrored on the other side to ensure the equality remains valid. This concept is not just crucial for this particular equation but is a cornerstone of solving all algebraic equations. Understanding and applying these principles correctly ensures accurate solutions and builds a strong foundation for more advanced mathematical concepts.
Real-World Applications
Solving equations like 45s + 16 = 151 isn't just an academic exercise; it has numerous real-world applications. These types of linear equations appear in various fields, such as physics, engineering, economics, and computer science. For example, in physics, you might use such an equation to calculate the time it takes for an object to travel a certain distance at a constant speed, accounting for an initial displacement. In engineering, these equations can help determine the stress on a material or the current in an electrical circuit.
In economics, linear equations are used in supply and demand models to find equilibrium prices and quantities. In computer science, they can be part of algorithms for optimization problems. The ability to solve these equations efficiently and accurately is a valuable skill in any quantitative field. Moreover, the logical thinking and problem-solving skills developed through algebra are transferable to many other areas of life, from financial planning to making informed decisions based on data. Therefore, mastering the basics of solving equations is not only beneficial for academic success but also for practical applications in various professions and everyday situations.
Conclusion
In conclusion, when solving the equation 45s + 16 = 151, the first two essential steps are to subtract 16 from both sides and then divide both sides by 45. These steps are critical because they systematically isolate the variable s, allowing us to find its value. Subtracting 16 removes the constant term, and dividing by 45 undoes the multiplication, leaving s by itself. These operations are grounded in the fundamental principles of algebraic manipulation, which require maintaining balance by performing the same actions on both sides of the equation.
Understanding these steps and the reasoning behind them is crucial for mastering algebra and solving more complex equations. It's also important to avoid common mistakes, such as performing operations in the wrong order or not applying the same operation to both sides. The ability to solve equations like this has numerous real-world applications in various fields, making it a valuable skill. By following these steps and practicing regularly, anyone can improve their equation-solving abilities and build a solid mathematical foundation.
