Solving 3x-2/2x+1=4/5 A Step-by-Step Guide
In the realm of algebra, solving equations is a fundamental skill. This article delves into the process of solving the equation 3x-2/2x+1=4/5, providing a step-by-step guide that is both comprehensive and easy to follow. Whether you're a student grappling with algebraic equations or simply someone looking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems.
Understanding the Equation
Before we jump into the solution, let's break down the equation 3x-2/2x+1=4/5. This is a rational equation, which means it involves fractions where the numerator and denominator are polynomials. Our goal is to find the value(s) of 'x' that make this equation true. To achieve this, we'll employ a series of algebraic manipulations, ensuring that each step maintains the equality of the equation. The core principle we'll use is to isolate 'x' on one side of the equation, revealing its value. This involves undoing the operations performed on 'x', such as addition, subtraction, multiplication, and division, in the reverse order. We'll also need to handle the fractions carefully, which will involve techniques like cross-multiplication. Remember, each step we take must be justified by a property of equality, ensuring that we're not changing the fundamental relationship expressed by the equation. So, let's embark on this algebraic journey and unravel the value of 'x'. Understanding the structure of the equation is crucial for selecting the appropriate solution strategy. The presence of fractions necessitates a method that eliminates them, which is precisely what we'll do in the following steps. By understanding the underlying principles, you'll not only solve this equation but also gain a deeper appreciation for the beauty and logic of algebra. This understanding will empower you to approach other equations with confidence, knowing that you have a solid foundation to build upon. The key is to approach each step methodically and to check your work to ensure accuracy. With practice, solving rational equations will become second nature, and you'll be able to tackle more complex algebraic challenges.
Step 1: Cross-Multiplication
The first step in solving the equation 3x-2/2x+1=4/5 is to eliminate the fractions. We achieve this by cross-multiplication. This technique involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. This effectively clears the denominators and transforms the equation into a more manageable form. When performing cross-multiplication, it's crucial to distribute the multiplication correctly. This means that each term in the numerator must be multiplied by the denominator, and vice versa. A common mistake is to only multiply the first term, which can lead to an incorrect solution. By carefully applying the distributive property, we ensure that the equality of the equation is maintained. Cross-multiplication is a powerful tool for solving rational equations, and it's a technique that you'll use frequently in algebra. It's important to understand why it works: we're essentially multiplying both sides of the equation by the product of the denominators, which cancels out the denominators on each side. This process simplifies the equation while preserving its solution. After cross-multiplication, we obtain a linear equation, which is much easier to solve. This is a significant step forward in our solution process. Remember, the goal of each step is to simplify the equation and bring us closer to isolating 'x'. By mastering cross-multiplication, you'll be well-equipped to handle a wide range of rational equations. This step is not just a mechanical procedure; it's a strategic move that sets the stage for the rest of the solution. So, let's proceed with confidence and continue our journey towards finding the value of 'x'.
Performing Cross-Multiplication
Cross-multiplying the equation 3x-2/2x+1=4/5 involves multiplying (3x - 2) by 5 and (2x + 1) by 4. This gives us the equation: 5(3x - 2) = 4(2x + 1). This step is crucial because it transforms the original equation, which involves fractions, into a simpler form that is easier to manipulate. The key here is to ensure that the multiplication is carried out correctly, paying close attention to the signs and the distribution of terms. The left side of the equation, 5(3x - 2), represents the product of 5 and the binomial (3x - 2). Similarly, the right side, 4(2x + 1), represents the product of 4 and the binomial (2x + 1). These products are equivalent due to the principle of cross-multiplication, which is a valid operation for equations involving fractions. By equating these products, we are essentially stating that the ratio of (3x - 2) to (2x + 1) is the same as the ratio of 4 to 5, which is what the original equation conveys. This transformation is a powerful technique in algebra, allowing us to move from a potentially complex fractional equation to a more straightforward linear equation. The next step will involve distributing the multiplication over the binomials, which will further simplify the equation and bring us closer to isolating the variable 'x'. Remember, each step in the process is designed to make the equation easier to solve while maintaining its original meaning. Careful execution of this step is paramount to achieving the correct solution. Let's move on to the next step and continue the journey towards finding the value of 'x'.
Step 2: Distribute
Following the cross-multiplication, we now have the equation 5(3x - 2) = 4(2x + 1). The next step is to distribute the constants on both sides of the equation. This involves multiplying the number outside the parentheses by each term inside the parentheses. Distribution is a fundamental algebraic operation, and it's essential to perform it accurately to maintain the integrity of the equation. A common mistake is to forget to multiply the constant by every term inside the parentheses, which can lead to an incorrect solution. By carefully applying the distributive property, we ensure that each term is multiplied correctly. This step simplifies the equation further by removing the parentheses and combining like terms. After distribution, we'll have a linear equation with 'x' terms and constant terms on both sides. This form is much easier to solve than the original equation with fractions. The distributive property is a cornerstone of algebra, and it's used extensively in solving equations and simplifying expressions. It's important to understand why it works: it's based on the idea that multiplication distributes over addition and subtraction. In other words, a(b + c) is the same as ab + ac, and a(b - c) is the same as ab - ac. By mastering the distributive property, you'll be well-equipped to handle a wide range of algebraic problems. This step is a crucial bridge between the cross-multiplication and the isolation of 'x'. So, let's proceed with confidence and distribute the constants, bringing us closer to our goal.
Applying the Distributive Property
Applying the distributive property to the equation 5(3x - 2) = 4(2x + 1), we multiply 5 by both 3x and -2 on the left side, and 4 by both 2x and 1 on the right side. This results in the equation 15x - 10 = 8x + 4. This step is a critical simplification, as it expands the equation and prepares it for further manipulation to isolate the variable 'x'. The distributive property is a cornerstone of algebraic operations, and its correct application here is essential for maintaining the equality of the equation. Each term within the parentheses is affected by the multiplication, ensuring that the relationship between the two sides remains balanced. The left side of the equation, initially 5(3x - 2), transforms into 15x - 10 through distribution. This means that 5 is multiplied by 3x, yielding 15x, and 5 is multiplied by -2, resulting in -10. Similarly, on the right side, 4(2x + 1) becomes 8x + 4, where 4 is multiplied by 2x to give 8x, and 4 is multiplied by 1 to give 4. This expansion reveals the individual terms that contribute to the overall equation, making it easier to identify like terms and proceed with the simplification process. The resulting equation, 15x - 10 = 8x + 4, is a linear equation in a more standard form, allowing us to use familiar techniques to solve for 'x'. This includes combining like terms and isolating the variable on one side of the equation. The accuracy of this step is paramount, as any error in the distribution will propagate through the rest of the solution. Let's move forward with the next step, continuing the process of isolating 'x' and finding its value.
Step 3: Combine Like Terms
After distributing, our equation is 15x - 10 = 8x + 4. Now, we need to combine like terms to further simplify the equation. This involves gathering all the 'x' terms on one side of the equation and all the constant terms on the other side. Combining like terms is a fundamental algebraic technique that helps to isolate the variable and make the equation easier to solve. To do this, we'll use the addition and subtraction properties of equality. This means we can add or subtract the same value from both sides of the equation without changing its solution. The goal is to move all the 'x' terms to one side and all the constants to the other side, which will allow us to isolate 'x'. When combining like terms, it's important to pay attention to the signs of the terms. A common mistake is to forget to change the sign when moving a term from one side of the equation to the other. By carefully applying the addition and subtraction properties of equality, we ensure that the equation remains balanced and that we're not changing its solution. This step is a crucial step in solving linear equations, and it's a technique that you'll use frequently in algebra. By mastering this step, you'll be well-equipped to handle a wide range of algebraic problems. Combining like terms simplifies the equation and brings us closer to isolating 'x'. So, let's proceed with confidence and combine the like terms, moving us closer to our goal.
Isolating x Terms
To isolate the 'x' terms in the equation 15x - 10 = 8x + 4, we subtract 8x from both sides. This gives us 15x - 8x - 10 = 8x - 8x + 4, which simplifies to 7x - 10 = 4. This step is a key maneuver in solving for 'x', as it consolidates all the 'x' terms on one side of the equation, paving the way for further isolation. Subtracting 8x from both sides maintains the equation's balance, a principle rooted in the properties of equality. What we do to one side, we must do to the other, ensuring that the relationship between the two sides remains consistent. The subtraction eliminates the 8x term on the right side, leaving only the constant term 4. On the left side, 15x - 8x combines to give 7x, while the -10 remains unchanged for now. The resulting equation, 7x - 10 = 4, is significantly simpler than the original, and it brings us closer to the ultimate goal of finding the value of 'x'. This process of isolating the variable is a fundamental technique in algebra, and it's applicable to a wide range of equations. It involves strategically using inverse operations to peel away the layers of terms and constants that surround the variable, eventually revealing its value. The accuracy of this step is critical, as any error in the subtraction will affect the subsequent steps and the final solution. Let's continue with the next step, further isolating 'x' and bringing us closer to the answer.
Isolating Constant Terms
With the equation now in the form 7x - 10 = 4, our next task is to isolate the constant terms. To do this, we add 10 to both sides of the equation. This yields 7x - 10 + 10 = 4 + 10, which simplifies to 7x = 14. This step is crucial because it moves all the constant terms to the right side of the equation, leaving only the term with 'x' on the left side. Adding 10 to both sides is a direct application of the addition property of equality, which states that adding the same value to both sides of an equation maintains the equality. This principle is fundamental to solving equations, as it allows us to manipulate terms without changing the solution. The addition effectively cancels out the -10 on the left side, leaving 7x by itself. On the right side, 4 + 10 combines to give 14. The resulting equation, 7x = 14, is a significant simplification, as it isolates the 'x' term and sets the stage for the final step of dividing to solve for 'x'. This process of isolating terms is a recurring theme in algebra, and it's a skill that becomes increasingly important as equations become more complex. By mastering this technique, you'll be well-equipped to tackle a wide range of algebraic challenges. The accuracy of this step is paramount, as any error in the addition will affect the final solution. Let's move on to the final step and complete the process of solving for 'x'.
Step 4: Divide
We've arrived at the equation 7x = 14. The final step in solving for 'x' is to divide both sides of the equation by the coefficient of 'x', which is 7 in this case. This step isolates 'x' and reveals its value. Division is the inverse operation of multiplication, and it's the key to undoing the multiplication of 7 and 'x'. By dividing both sides by 7, we're essentially asking:
