Solving (3/4)x + 3 - 2x = (-1/4) + (1/2)x + 5 A Step-by-Step Guide

Solving linear equations is a fundamental skill in mathematics, serving as a cornerstone for more advanced topics. This article will provide a comprehensive guide to solving the equation $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$, walking you through each step with clarity and precision. Understanding the underlying principles will empower you to tackle a wide range of algebraic problems with confidence. This article is meticulously crafted to enhance your understanding of algebraic manipulations, focusing on the techniques required to isolate variables and find solutions. Whether you're a student grappling with homework or someone looking to brush up on your math skills, this guide offers a clear and structured approach to mastering linear equations.

The ability to solve equations is not just an academic exercise; it's a crucial skill in various real-world scenarios, from budgeting and finance to engineering and scientific calculations. By understanding the mechanics of equation solving, you equip yourself with a powerful tool for problem-solving in both abstract and practical contexts. The equation we're addressing, $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$, might seem daunting at first, but with a systematic approach, it becomes manageable. This guide breaks down the process into distinct stages, ensuring you grasp each concept before moving on to the next. This step-by-step method will help you build a solid foundation for tackling more complex mathematical problems in the future.

Our primary goal is to isolate the variable 'x' on one side of the equation, which will reveal its value. This involves a series of algebraic manipulations, including combining like terms, eliminating fractions, and performing operations on both sides of the equation to maintain balance. Each step is carefully explained, emphasizing the logic behind the manipulation to ensure not just the ability to solve this specific equation, but also the understanding to solve similar problems. By the end of this guide, you'll not only know the solution to this equation but also understand the process and principles involved in solving linear equations in general. This knowledge will be invaluable as you progress in your mathematical journey, allowing you to approach new challenges with confidence and competence.

1. Clearing Fractions: Multiplying to Simplify

The initial hurdle in solving the equation $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$ lies in the presence of fractions. Fractions can complicate algebraic manipulations, making it essential to eliminate them early in the solving process. The most effective way to clear fractions is by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 4, 1 (for the whole numbers), and 2. The LCM of 4 and 2 is 4, making it the ideal multiplier for simplifying the equation. This step is crucial because it transforms the equation into a more manageable form, removing the fractional coefficients and paving the way for further simplification.

When we multiply both sides of the equation by 4, we apply the distributive property to each term. This means that each fraction and whole number is multiplied by 4, ensuring that the equation remains balanced. The process involves multiplying 4 by $\frac{3}{4}x$, 3, -2x, -$\frac{1}{4}$, $\frac{1}{2}x$, and 5. This careful distribution is key to preserving the equation's integrity and ensuring that the solution we arrive at is accurate. The goal is to eliminate the fractions without altering the fundamental relationship between the terms in the equation. This transformation is a critical step in simplifying the equation and making it easier to solve.

Multiplying each term by 4 results in the following transformation: $(4 * \frac{3}{4}x) + (4 * 3) - (4 * 2x) = (4 * -\frac{1}{4}) + (4 * \frac{1}{2}x) + (4 * 5)$. Simplifying each multiplication, we get 3x + 12 - 8x = -1 + 2x + 20. This new form of the equation is free of fractions, making it significantly easier to work with. The coefficients are now whole numbers, which simplifies the subsequent steps of combining like terms and isolating the variable. This process highlights the power of using the LCM to clear fractions, a technique that is applicable to a wide range of algebraic equations. This simplification is a major step forward in solving the equation, setting the stage for further algebraic manipulations.

2. Combining Like Terms: Simplifying Expressions

After clearing the fractions, the next step in solving the equation is to combine like terms on each side. This process involves grouping terms that have the same variable (in this case, 'x') and combining constant terms. Like terms are terms that have the same variable raised to the same power. By combining like terms, we simplify the equation, reducing the number of terms and making it easier to isolate the variable. This simplification is a crucial step in solving the equation efficiently and accurately.

On the left side of the equation, 3x and -8x are like terms, and on the right side, -1 and 20 are like terms. Combining 3x and -8x yields -5x, while combining -1 and 20 results in 19. This process of combining like terms reduces the complexity of the equation, making it more manageable. The equation now becomes -5x + 12 = 2x + 19. This simplified form is much easier to work with, allowing us to focus on isolating the variable 'x'.

The act of combining like terms is essentially an application of the distributive property in reverse. It involves adding or subtracting the coefficients of the like terms while keeping the variable unchanged. This step is fundamental to algebraic simplification and is used extensively in solving equations. By reducing the number of terms, we make the equation more concise and easier to manipulate. This process also helps to reveal the underlying structure of the equation, making it clearer how to proceed with the next steps. The simplified equation, -5x + 12 = 2x + 19, represents a significant step towards isolating 'x' and finding the solution.

3. Isolating the Variable: Moving Terms Across the Equality

Isolating the variable is the heart of solving any equation. In the equation -5x + 12 = 2x + 19, our goal is to get all terms containing 'x' on one side and all constant terms on the other. This process involves adding or subtracting terms from both sides of the equation to maintain balance. The principle at play here is that any operation performed on one side of the equation must also be performed on the other side to preserve equality. This step is crucial for bringing the equation closer to its solution.

To begin isolating 'x', we can add 5x to both sides of the equation. This will eliminate the -5x term on the left side and move the 'x' terms to the right side. Adding 5x to both sides results in 12 = 2x + 5x + 19, which simplifies to 12 = 7x + 19. This manipulation brings all the 'x' terms to one side, making it easier to isolate 'x'. This is a common technique in solving equations, and understanding it is essential for algebraic proficiency.

Next, we need to move the constant term from the right side to the left side. To do this, we subtract 19 from both sides of the equation. This operation eliminates the +19 on the right side and moves the constant term to the left side. Subtracting 19 from both sides gives us 12 - 19 = 7x, which simplifies to -7 = 7x. This equation is now in a form where it's straightforward to isolate 'x'. The process of adding and subtracting terms from both sides is a fundamental skill in algebra, and mastering it is key to solving equations effectively. This careful manipulation is essential to finding the correct solution.

4. Solving for x: Dividing to Find the Solution

With the equation simplified to -7 = 7x, the final step in solving for x is to divide both sides by the coefficient of 'x', which in this case is 7. This isolates 'x' on one side of the equation, revealing its value. The principle behind this step is that dividing both sides of the equation by the same non-zero number maintains the equality. This is the final algebraic manipulation needed to find the solution.

Dividing both sides of the equation -7 = 7x by 7, we get -7/7 = (7x)/7. This simplifies to -1 = x. Therefore, the solution to the equation is x = -1. This solution is the value of 'x' that makes the original equation true. The process of dividing to isolate the variable is a common final step in solving equations, and it's crucial for finding the correct solution.

The solution, x = -1, is the value that satisfies the original equation $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$. This means that if we substitute -1 for 'x' in the original equation, both sides will be equal. This is a way to check our work and ensure that we have arrived at the correct solution. The ability to solve for 'x' in this manner is a fundamental skill in algebra, and it's used extensively in various mathematical contexts.

5. Verification: Checking the Solution

After finding a solution to an equation, verification is a crucial step to ensure accuracy. This involves substituting the solution back into the original equation to confirm that it satisfies the equation. This step is essential to catch any errors made during the solving process and to build confidence in the solution. Verifying the solution is a fundamental practice in mathematics, ensuring that the answer is correct.

To verify the solution x = -1, we substitute it into the original equation $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$. Substituting -1 for 'x' gives us $\frac{3}{4}(-1) + 3 - 2(-1) = -\frac{1}{4} + \frac{1}{2}(-1) + 5$. Now, we simplify both sides of the equation.

On the left side, $\frac{3}{4}(-1)$ is -$\frac{3}{4}$, -2(-1) is 2, so the left side becomes -$\frac{3}{4}$ + 3 + 2. Converting the whole numbers to fractions with a denominator of 4, we get -$\frac{3}{4}$ + $\frac{12}{4}$ + $\frac{8}{4}$, which simplifies to $\frac{17}{4}$. On the right side, $\frac{1}{2}(-1)$ is -$\frac{1}{2}$, so the right side becomes -$\frac{1}{4}$ - $\frac{1}{2}$ + 5. Converting the fractions to a common denominator of 4, we get -$\frac{1}{4}$ - $\frac{2}{4}$ + $\frac{20}{4}$, which simplifies to $\frac{17}{4}$.

Since both sides of the equation simplify to $\frac{17}{4}$, the solution x = -1 is verified. This confirms that our solution is correct and that the algebraic manipulations we performed were accurate. Verification is a valuable step in the problem-solving process, providing assurance that the answer is correct. This thorough check solidifies the understanding of the equation and its solution.

In conclusion, mastering equation solving is a crucial skill in mathematics that empowers you to tackle a wide range of problems. This guide has provided a step-by-step approach to solving the equation $\frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5$, emphasizing the underlying principles and techniques involved. By understanding how to clear fractions, combine like terms, isolate variables, and verify solutions, you can confidently approach various algebraic equations. The ability to solve equations is not just an academic exercise but a valuable tool in real-world applications, from budgeting and finance to science and engineering. This comprehensive guide equips you with the knowledge and skills needed to excel in mathematics and beyond.

The process of solving equations involves a series of algebraic manipulations designed to isolate the variable. Each step is performed with the goal of simplifying the equation and bringing it closer to its solution. The steps we've covered—clearing fractions, combining like terms, isolating the variable, solving for 'x', and verifying the solution—are fundamental to solving linear equations. By practicing these techniques, you can develop fluency in algebraic problem-solving. This proficiency will serve as a solid foundation for more advanced mathematical concepts.

The solution to the equation, x = -1, represents the value that makes the equation true. Verification is a crucial step in the problem-solving process, ensuring that the solution is accurate. By substituting the solution back into the original equation, we confirm that both sides are equal, providing confidence in our answer. This practice of verification reinforces the understanding of equations and their solutions. The skills and knowledge gained from this guide will empower you to tackle a variety of mathematical challenges with confidence and competence.