Solving Linear Equations Step-by-Step Solutions For Q2 1 And 2
In the realm of mathematics, solving linear equations stands as a fundamental skill. It's the cornerstone upon which more advanced algebraic concepts are built. Linear equations, characterized by variables raised to the power of one, pop up in various real-world scenarios, making their mastery incredibly important. In this comprehensive guide, we will solve two linear equations step-by-step, ensuring that you grasp not just the solutions, but also the underlying principles and methodologies involved.
Q2 (1): Deciphering and Solving (x/2) - (1/5) = (x/3) + (1/4)
Our journey begins with the first equation: (x/2) - (1/5) = (x/3) + (1/4). This equation, while seemingly complex at first glance, is a classic example of a linear equation with fractions. To effectively solve this, we will employ a systematic approach, eliminating fractions and isolating the variable. Linear equations like this are the building blocks of algebra, and mastering them opens doors to more complex mathematical problem-solving.
Step 1: Eliminating Fractions Understanding the Least Common Multiple (LCM)
The presence of fractions can often complicate the solving process. To simplify matters, our initial step involves eliminating these fractions. This is achieved by multiplying both sides of the equation by the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of all the denominators in the equation. In our case, the denominators are 2, 5, 3, and 4. To find the LCM, we identify the prime factors of each denominator:
- 2 = 2
- 5 = 5
- 3 = 3
- 4 = 2 x 2
The LCM is then the product of the highest powers of all prime factors present: 2² x 3 x 5 = 60. Therefore, 60 is the LCM of 2, 5, 3, and 4. This seemingly simple step is crucial; by finding the LCM, we set the stage for a cleaner equation, free from the complexities of fractions.
Step 2: Multiplying by the LCM Clearing the Path
With the LCM identified as 60, we now multiply both sides of the equation by this value. This action is a pivotal moment in solving the equation, as it elegantly eliminates the fractions, transforming the equation into a more manageable form. Each term in the equation is multiplied by 60, effectively clearing the denominators. This process not only simplifies the equation but also brings us closer to isolating the variable, which is the ultimate goal in solving any linear equation.
Multiplying both sides of the equation (x/2) - (1/5) = (x/3) + (1/4) by 60 gives us:
60 * (x/2) - 60 * (1/5) = 60 * (x/3) + 60 * (1/4)
This simplifies to:
30x - 12 = 20x + 15
This transformation is not just about simplification; it's about strategic problem-solving. By clearing the fractions, we've paved the way for a more straightforward algebraic manipulation, bringing clarity and ease to the process of finding the value of 'x'.
Step 3: Rearranging Terms Isolating the Variable
Now that we have a simplified equation without fractions, our next step is to rearrange the terms. This involves grouping the terms containing 'x' on one side of the equation and the constant terms on the other side. This rearrangement is a fundamental technique in algebra, allowing us to isolate the variable and ultimately solve for its value. The process requires careful manipulation of the equation, ensuring that the balance is maintained by performing the same operations on both sides.
To achieve this, we subtract 20x from both sides of the equation:
30x - 20x - 12 = 20x - 20x + 15
This simplifies to:
10x - 12 = 15
Next, we add 12 to both sides:
10x - 12 + 12 = 15 + 12
Which simplifies to:
10x = 27
By strategically moving terms around, we've successfully isolated the 'x' term, bringing us one step closer to the final solution. This step is a testament to the power of algebraic manipulation in simplifying complex equations.
Step 4: Solving for x The Final Act
Having successfully isolated the term with 'x', the final step is to solve for 'x'. This is achieved by dividing both sides of the equation by the coefficient of 'x'. In our case, the coefficient of 'x' is 10. This division is the concluding act in our algebraic journey, revealing the value of 'x' that satisfies the original equation. It's a moment of culmination, where all the previous steps come together to provide a definitive answer.
Dividing both sides of the equation 10x = 27 by 10, we get:
x = 27/10
Therefore, the solution to the equation (x/2) - (1/5) = (x/3) + (1/4) is x = 2.7. This result is not just a number; it's the culmination of a methodical process, a testament to the power of algebraic principles in solving linear equations. The value of 'x' we've found is the unique solution that makes the equation true, a satisfying conclusion to our mathematical endeavor.
Q2 (2): Navigating x + 7 - 8x = (17/6) - (5x/2)
Our attention now turns to the second equation: x + 7 - 8x = (17/6) - (5x/2). This equation, like the first, presents a blend of integers, fractions, and variables. However, it offers a slightly different challenge, requiring us to first simplify the equation by combining like terms before proceeding with the elimination of fractions. This equation is a practical exercise in applying algebraic techniques to real-world problems, highlighting the importance of order and precision in mathematical calculations.
Step 1: Combining Like Terms Simplifying the Equation
Before we tackle the fractions in the equation, it's crucial to simplify it by combining like terms. Like terms are those that contain the same variable raised to the same power. In our equation, 'x' and '-8x' are like terms, and they can be combined to simplify the left side of the equation. This step is a fundamental practice in algebra, streamlining the equation and making it easier to work with. By combining like terms, we reduce the complexity of the equation, paving the way for a more straightforward solution.
Combining 'x' and '-8x' gives us:
x - 8x = -7x
So, the equation x + 7 - 8x = (17/6) - (5x/2) simplifies to:
-7x + 7 = (17/6) - (5x/2)
This simplification is not just about aesthetics; it's about efficiency. By reducing the number of terms, we make the equation more manageable, setting the stage for the subsequent steps in solving for 'x'.
Step 2: Eliminating Fractions Multiplying by the LCM
With the equation simplified, our next task is to eliminate the fractions. As we did in the first equation, we achieve this by multiplying both sides of the equation by the Least Common Multiple (LCM) of the denominators. In this case, the denominators are 6 and 2. The LCM of 6 and 2 is 6. This step is crucial for transforming the equation into a more workable form, free from the constraints of fractional coefficients.
Multiplying both sides of the equation -7x + 7 = (17/6) - (5x/2) by 6 gives us:
6 * (-7x + 7) = 6 * ((17/6) - (5x/2))
This expands to:
-42x + 42 = 17 - 15x
This transformation is a pivotal moment in our problem-solving journey. By eliminating the fractions, we've created a cleaner, more streamlined equation, making it easier to isolate the variable and find its value. This step underscores the importance of strategic manipulation in solving algebraic equations.
Step 3: Rearranging Terms Isolating x
Now that we have an equation without fractions, our focus shifts to rearranging the terms to isolate 'x'. This involves moving all terms containing 'x' to one side of the equation and all constant terms to the other side. This rearrangement is a key technique in algebra, allowing us to separate the variable from the constants, bringing us closer to the solution. The process requires careful attention to detail, ensuring that the equation remains balanced by performing the same operations on both sides.
To begin, we add 15x to both sides of the equation:
-42x + 15x + 42 = 17 - 15x + 15x
This simplifies to:
-27x + 42 = 17
Next, we subtract 42 from both sides:
-27x + 42 - 42 = 17 - 42
Which simplifies to:
-27x = -25
By strategically moving terms, we've successfully isolated the 'x' term, setting the stage for the final step in solving for 'x'. This rearrangement is a testament to the power of algebraic manipulation in simplifying complex equations.
Step 4: Solving for x The Grand Finale
With the 'x' term isolated, the final step is to solve for 'x'. This is achieved by dividing both sides of the equation by the coefficient of 'x'. In our case, the coefficient of 'x' is -27. This division is the concluding act in our algebraic drama, revealing the value of 'x' that satisfies the original equation. It's a moment of triumph, where all the preceding steps culminate in a definitive answer.
Dividing both sides of the equation -27x = -25 by -27, we get:
x = (-25) / (-27)
Which simplifies to:
x = 25/27
Therefore, the solution to the equation x + 7 - 8x = (17/6) - (5x/2) is x = 25/27. This result is not just a number; it's the outcome of a meticulous process, a testament to the elegance and precision of algebra. The value of 'x' we've found is the unique solution that makes the equation true, a satisfying conclusion to our mathematical exploration.
Conclusion: Mastering Linear Equations A Gateway to Advanced Mathematics
In this guide, we've meticulously solved two linear equations, each presenting its unique challenges and requiring a strategic approach. From eliminating fractions to combining like terms and isolating the variable, we've navigated the intricacies of algebraic manipulation. These techniques are not just tools for solving equations; they are fundamental skills that pave the way for understanding more advanced mathematical concepts.
The ability to solve linear equations is a cornerstone of mathematical literacy. It empowers you to tackle a wide range of problems, both in academic settings and real-world scenarios. As you continue your mathematical journey, the skills you've honed here will serve as a solid foundation, enabling you to explore the fascinating world of algebra and beyond. So, embrace the challenge, practice diligently, and unlock the power of linear equations in your mathematical pursuits.
