Solving 4y - 8y + 3 = 0 A Step-by-Step Guide
In this comprehensive guide, we will walk you through the process of solving the equation 4y - 8y + 3 = 0 step-by-step. This type of equation is a linear equation, and understanding how to solve it is a fundamental skill in algebra. Whether you're a student tackling homework or just brushing up on your math skills, this guide will provide a clear and easy-to-follow explanation. We'll cover the key concepts, the necessary steps, and offer additional tips to ensure you grasp the process thoroughly. So, let's dive in and solve this equation together!
Understanding Linear Equations
Before we start solving the equation 4y - 8y + 3 = 0, it’s crucial to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is raised to the power of 1, and there are no exponents or other complex functions involved. Linear equations are fundamental in mathematics and appear in various real-world applications, from simple calculations to complex scientific models. They are characterized by their straightforward structure and the fact that their graphs are straight lines.
The general form of a linear equation is ax + b = 0, where 'x' represents the variable, and 'a' and 'b' are constants. The goal in solving a linear equation is to isolate the variable on one side of the equation to find its value. This involves using algebraic manipulations such as addition, subtraction, multiplication, and division while maintaining the equation's balance. Understanding the properties of equality is key to this process. The addition property of equality states that adding the same value to both sides of an equation does not change the solution. Similarly, the subtraction, multiplication, and division properties of equality allow us to perform these operations on both sides of the equation without altering its balance. By applying these properties systematically, we can simplify the equation and isolate the variable.
Linear equations can have one solution, no solution, or infinitely many solutions. An equation with one solution means that there is a single value for the variable that makes the equation true. For example, the equation x + 2 = 5 has one solution, x = 3. An equation with no solution, also known as a contradiction, occurs when the equation simplifies to a false statement, such as 0 = 1. In this case, there is no value for the variable that can satisfy the equation. Finally, an equation with infinitely many solutions, also known as an identity, simplifies to a true statement, such as 0 = 0. This means that any value for the variable will make the equation true. Recognizing the type of solution an equation has is an important part of the problem-solving process.
Step 1: Combine Like Terms
The first step in solving the equation 4y - 8y + 3 = 0 is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In this equation, 4y and -8y are like terms because they both contain the variable y raised to the power of 1. Combining these terms simplifies the equation and makes it easier to solve. This step is crucial in reducing the complexity of the equation and setting the stage for isolating the variable.
To combine like terms, you simply add or subtract their coefficients. The coefficient is the numerical factor in a term. In the term 4y, the coefficient is 4, and in the term -8y, the coefficient is -8. So, to combine 4y and -8y, we add their coefficients: 4 + (-8) = -4. This means that 4y - 8y simplifies to -4y. The simplified equation now becomes -4y + 3 = 0. This process of combining like terms is based on the distributive property of multiplication over addition, which allows us to factor out the common variable and simplify the expression.
Combining like terms is a fundamental skill in algebra and is used extensively in solving various types of equations, not just linear equations. It is important to pay close attention to the signs (positive or negative) of the coefficients when combining like terms. A common mistake is to overlook the negative sign, which can lead to an incorrect solution. By carefully combining like terms, we can reduce the equation to a simpler form, making the subsequent steps of solving the equation much easier. This step not only simplifies the equation but also prepares it for further algebraic manipulation to isolate the variable.
Step 2: Isolate the Variable Term
After combining like terms, the equation 4y - 8y + 3 = 0 has been simplified to -4y + 3 = 0. The next step is to isolate the variable term, which in this case is -4y. To isolate the variable term, we need to eliminate any constants that are on the same side of the equation. In this case, we have the constant +3 on the left side of the equation. Isolating the variable term is a critical step because it brings us closer to determining the value of the variable itself.
To eliminate the constant +3, we use the subtraction property of equality, which states that subtracting the same value from both sides of an equation does not change the solution. We subtract 3 from both sides of the equation: -4y + 3 - 3 = 0 - 3. This simplifies to -4y = -3. By subtracting 3 from both sides, we have successfully removed the constant term from the left side of the equation, leaving the variable term -4y isolated. This step is based on the principle of maintaining balance in the equation, ensuring that whatever operation is performed on one side is also performed on the other side.
Isolating the variable term is a common technique used in solving algebraic equations. It is important to perform the operation correctly and to both sides of the equation to maintain its balance. A common mistake is to forget to perform the operation on both sides, which can lead to an incorrect solution. By carefully isolating the variable term, we set up the equation for the final step of solving for the variable. This step ensures that we are one step closer to finding the value of 'y' that satisfies the original equation, making it a crucial part of the problem-solving process.
Step 3: Solve for the Variable
Now that we have isolated the variable term, the equation is -4y = -3. The final step is to solve for the variable y. To do this, we need to get y by itself on one side of the equation. In this case, y is being multiplied by -4. To undo this multiplication, we use the division property of equality, which states that dividing both sides of an equation by the same non-zero value does not change the solution. Solving for the variable is the culmination of the previous steps and provides the answer to the equation.
To isolate y, we divide both sides of the equation by -4: (-4y) / (-4) = (-3) / (-4). This simplifies to y = 3/4. When we divide -4y by -4, the -4s cancel out, leaving us with y. On the other side of the equation, when we divide -3 by -4, the negative signs cancel out, and we are left with 3/4. This means that the solution to the equation 4y - 8y + 3 = 0 is y = 3/4. This step is based on the principle of inverse operations, where we use the opposite operation to undo the operation being performed on the variable.
Solving for the variable is the ultimate goal in solving algebraic equations. It is important to perform the division correctly and to both sides of the equation to maintain its balance. A common mistake is to forget to divide both sides, which can lead to an incorrect solution. By carefully solving for the variable, we determine the value that satisfies the original equation. In this case, y = 3/4 is the value that makes the equation 4y - 8y + 3 = 0 true. This final step completes the process of solving the equation, providing a clear and concise solution.
Verification (Optional but Recommended)
After solving for the variable, it’s a good practice to verify your solution. This step is optional but highly recommended as it helps ensure that the value you found for the variable is correct. Verification involves substituting the solution back into the original equation to see if it makes the equation true. This process helps catch any errors that may have been made during the solving process and provides confidence in the correctness of the solution.
To verify the solution y = 3/4, we substitute this value back into the original equation 4y - 8y + 3 = 0. The equation becomes 4(3/4) - 8(3/4) + 3 = 0. First, we multiply: 4 * (3/4) = 3 and 8 * (3/4) = 6. So, the equation becomes 3 - 6 + 3 = 0. Now, we simplify: 3 - 6 = -3, and then -3 + 3 = 0. The equation simplifies to 0 = 0, which is a true statement. This confirms that our solution y = 3/4 is correct. Verification is a powerful tool in algebra and helps reinforce the understanding of the equation-solving process.
Verifying the solution not only confirms the correctness of the answer but also helps solidify the understanding of the algebraic manipulations used to solve the equation. It provides an opportunity to review each step and ensure that no mistakes were made. This step is particularly useful in exam situations where accuracy is crucial. By taking the time to verify the solution, you can increase your confidence in your problem-solving abilities and ensure that you are submitting correct answers. Verification is a final check that ensures the solution is accurate and the algebraic process was followed correctly.
Conclusion
In conclusion, solving the equation 4y - 8y + 3 = 0 involves several key steps: combining like terms, isolating the variable term, and solving for the variable. We began by combining 4y and -8y to simplify the equation to -4y + 3 = 0. Then, we isolated the variable term by subtracting 3 from both sides, resulting in -4y = -3. Finally, we solved for y by dividing both sides by -4, which gave us the solution y = 3/4. We also discussed the importance of verifying the solution by substituting it back into the original equation to ensure its correctness. Understanding these steps and practicing them will help you solve a variety of linear equations with confidence.
The ability to solve linear equations is a fundamental skill in mathematics and is essential for further studies in algebra and related fields. Linear equations appear in various real-world applications, making their mastery crucial for problem-solving in different contexts. By following the step-by-step approach outlined in this guide, you can systematically solve linear equations and avoid common mistakes. Remember to always combine like terms, isolate the variable term, and solve for the variable. Additionally, verifying your solution is a valuable practice that ensures accuracy and enhances your understanding of the equation-solving process.
Mastering linear equations provides a solid foundation for tackling more complex mathematical problems. The techniques used in solving linear equations, such as combining like terms and using inverse operations, are applicable to a wide range of algebraic equations. With practice, you can become proficient in solving linear equations and build confidence in your mathematical abilities. This step-by-step guide serves as a valuable resource for students and anyone looking to improve their algebra skills. By understanding the underlying principles and following the outlined steps, you can successfully solve linear equations and apply these skills to various mathematical challenges.
