Sarita's Solution Step-by-Step Guide To Solving The Equation 5(x-3) + 7(x+4) = 73
Unveiling Sarita's Method for Solving the Equation
In this article, we will delve into Sarita's comprehensive approach to solving the equation 5(x-3) + 7(x+4) = 73. This equation, a classic example of a linear equation with parentheses, requires a systematic method to arrive at the correct solution. Sarita's step-by-step solution provides a clear and concise roadmap for navigating such problems. We will not only present Sarita's solution but also break down each step, offering explanations and insights into the underlying mathematical principles. This detailed analysis will be beneficial for students learning algebra, individuals seeking to refresh their equation-solving skills, and anyone interested in understanding the beauty and logic of mathematics.
The Importance of a Structured Approach
Solving algebraic equations, particularly those involving parentheses and multiple terms, necessitates a structured and methodical approach. Attempting to solve the equation haphazardly can easily lead to errors and confusion. Sarita's solution exemplifies the importance of following a logical sequence of steps. By meticulously addressing each operation and simplification, she minimizes the risk of mistakes and ensures a clear path towards the final answer. This structured approach is not just crucial for accuracy; it also fosters a deeper understanding of the mathematical concepts involved. By understanding the "why" behind each step, students can develop a more robust and adaptable problem-solving skill set.
Deciphering the Steps: A Detailed Breakdown
Sarita's solution likely involves the following key steps, which we will explore in detail: First, Distributing the constants outside the parentheses. This involves multiplying the constant by each term inside the parentheses. Second, Combining like terms. After distributing, we will have several terms involving 'x' and constant terms. Combining like terms simplifies the equation. Third, Isolating the variable. This involves moving all terms containing 'x' to one side of the equation and all constant terms to the other side. Fourth, Solving for x. Finally, we will divide both sides of the equation by the coefficient of 'x' to obtain the solution.
Sarita's Solution: A Step-by-Step Guide
Let's examine Sarita's solution step by step, providing detailed explanations and justifications for each operation. This section will serve as a practical guide for understanding the mechanics of solving the equation and will highlight the underlying algebraic principles at play.
Step 1: Distribution
The initial step in Sarita's solution involves distributing the constants outside the parentheses. This is a crucial step that eliminates the parentheses and allows us to simplify the equation. To distribute, we multiply the constant outside the parentheses by each term inside the parentheses.
For the first term, 5(x-3), we multiply 5 by both 'x' and '-3'. This gives us 5 * x = 5x and 5 * -3 = -15. So, 5(x-3) becomes 5x - 15.
Similarly, for the second term, 7(x+4), we multiply 7 by both 'x' and '4'. This gives us 7 * x = 7x and 7 * 4 = 28. Therefore, 7(x+4) becomes 7x + 28.
After distributing, the equation 5(x-3) + 7(x+4) = 73 transforms into 5x - 15 + 7x + 28 = 73. This step is fundamental because it removes the parentheses, making the equation easier to manipulate and solve. The distributive property, which justifies this step, is a cornerstone of algebra and is essential for simplifying expressions and equations.
Step 2: Combining Like Terms
Following the distribution step, the next logical step in Sarita's solution is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our equation, 5x - 15 + 7x + 28 = 73, the like terms are 5x and 7x (both containing the variable 'x') and -15 and 28 (both constant terms).
To combine the 'x' terms, we add their coefficients: 5x + 7x = (5 + 7)x = 12x.
To combine the constant terms, we add them: -15 + 28 = 13.
After combining like terms, the equation 5x - 15 + 7x + 28 = 73 simplifies to 12x + 13 = 73. This step is crucial because it reduces the number of terms in the equation, making it more manageable. By combining like terms, we are essentially simplifying the expression on the left side of the equation, paving the way for isolating the variable.
Step 3: Isolating the Variable
Once we have combined like terms, Sarita's next step likely involves isolating the variable. Isolating the variable means getting the term containing 'x' by itself on one side of the equation. In our simplified equation, 12x + 13 = 73, we need to isolate the term 12x.
To do this, we need to eliminate the constant term, 13, from the left side of the equation. We can achieve this by subtracting 13 from both sides of the equation. This is based on the principle that performing the same operation on both sides of an equation maintains the equality.
Subtracting 13 from both sides gives us: 12x + 13 - 13 = 73 - 13, which simplifies to 12x = 60. This step is pivotal because it brings us closer to solving for 'x'. By isolating the variable term, we are effectively setting the stage for the final step, where we will determine the value of 'x'.
Step 4: Solving for x
The final step in Sarita's solution is to solve for x. After isolating the variable term, we have the equation 12x = 60. To find the value of 'x', we need to undo the multiplication by 12.
We can do this by dividing both sides of the equation by 12. Again, this is based on the principle of maintaining equality by performing the same operation on both sides.
Dividing both sides by 12 gives us: 12x / 12 = 60 / 12, which simplifies to x = 5. Therefore, the solution to the equation 5(x-3) + 7(x+4) = 73 is x = 5. This final step completes the solution process, providing us with the numerical value of the unknown variable. The ability to solve for 'x' is a fundamental skill in algebra and is essential for tackling a wide range of mathematical problems.
Discussion: Why This Approach Works
Sarita's approach to solving the equation is rooted in fundamental algebraic principles. Let's delve into the reasons why this method is effective and mathematically sound.
The Foundation of Algebraic Principles
The success of Sarita's solution hinges on the application of core algebraic principles, particularly the distributive property and the properties of equality. The distributive property allows us to eliminate parentheses by multiplying a constant by each term within the parentheses. This step is crucial for simplifying the equation and making it easier to manipulate. The properties of equality dictate that performing the same operation on both sides of an equation maintains the balance and validity of the equation. This principle is applied when we subtract 13 from both sides and when we divide both sides by 12. These properties ensure that each step we take in solving the equation is mathematically sound and preserves the solution.
The Logic of Isolating the Variable
The strategy of isolating the variable is a cornerstone of equation-solving techniques. By isolating 'x', we are essentially stripping away all the other terms and operations that are obscuring its value. This process involves systematically undoing the operations that have been applied to 'x'. For example, if 'x' is being multiplied by 12, we undo this by dividing by 12. Similarly, if a constant is being added to the term containing 'x', we undo this by subtracting the constant. This systematic undoing of operations is a powerful technique that allows us to reveal the value of the unknown variable.
The Importance of Order of Operations
While not explicitly stated, Sarita's solution implicitly follows the order of operations (often remembered by the acronym PEMDAS/BODMAS), which dictates the sequence in which mathematical operations should be performed. This order ensures consistency and clarity in mathematical calculations. In this case, the distribution step comes before combining like terms, which in turn comes before isolating the variable. Adhering to the order of operations is crucial for arriving at the correct solution and avoiding errors.
Conclusion: Mastering Equation Solving
Sarita's solution provides a clear and concise demonstration of how to solve the equation 5(x-3) + 7(x+4) = 73. By understanding the underlying principles and following a structured approach, students and individuals can confidently tackle similar algebraic equations. The key takeaways from this analysis include the importance of the distributive property, combining like terms, isolating the variable, and adhering to the order of operations. Mastering these concepts is essential for building a strong foundation in algebra and for success in higher-level mathematics. This step-by-step approach not only helps in solving this specific equation but also equips learners with a problem-solving framework that can be applied to a wide range of mathematical challenges. The ability to solve equations is a fundamental skill that has applications in various fields, from science and engineering to finance and economics. By mastering these techniques, individuals can unlock a world of opportunities and enhance their analytical and problem-solving abilities.
