Solving 3(-y+7)=3(y+5)+6 Equation A Comprehensive Guide

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In this article, we will delve into the process of solving the equation $3(-y+7)=3(y+5)+6$. We will explore each step meticulously to arrive at the correct solution and understand the underlying mathematical principles. This comprehensive guide aims to provide clarity and insight into the equation-solving process, making it accessible for learners of all levels. Our primary focus will be on determining which of the following statements is true about the given equation:

A. The equation has one solution, $y=0$. B. The equation has one solution, $y=-1$. C. The equation has no solution. D. The equation has infinitely many solutions.

To accurately answer this question, we will systematically simplify the equation, isolate the variable y, and analyze the result. This step-by-step approach will not only help us identify the correct answer but also reinforce the fundamental concepts of algebraic manipulation.

Step-by-Step Solution

Let's begin by addressing the original equation: $3(-y+7)=3(y+5)+6$.

1. Distribute the Constants

The first step in solving this equation is to distribute the constants on both sides of the equation. This involves multiplying the numbers outside the parentheses by each term inside the parentheses.

On the left side, we have $3(-y+7)$. Distributing the 3, we get:

3∗−y+3∗7=−3y+213 * -y + 3 * 7 = -3y + 21

On the right side, we have $3(y+5)+6$. Distributing the 3 to the terms inside the parentheses, we get:

3∗y+3∗5=3y+153 * y + 3 * 5 = 3y + 15

Now, we add the +6 that was initially outside the parentheses:

3y+15+6=3y+213y + 15 + 6 = 3y + 21

So, after distributing the constants, our equation becomes:

−3y+21=3y+21-3y + 21 = 3y + 21

2. Combine Like Terms

The next step is to combine like terms. In this case, we want to group the y terms on one side of the equation and the constant terms on the other side. To do this, we can add $3y$ to both sides of the equation to eliminate the y term on the left side:

−3y+21+3y=3y+21+3y-3y + 21 + 3y = 3y + 21 + 3y

This simplifies to:

21=6y+2121 = 6y + 21

Now, we want to isolate the y term, so we subtract 21 from both sides of the equation:

21−21=6y+21−2121 - 21 = 6y + 21 - 21

This simplifies to:

0=6y0 = 6y

3. Isolate the Variable y

To isolate y, we need to divide both sides of the equation by the coefficient of y, which is 6:

06=6y6\frac{0}{6} = \frac{6y}{6}

This simplifies to:

0=y0 = y

So, we find that $y = 0$.

4. Verify the Solution

To ensure our solution is correct, we can substitute $y = 0$ back into the original equation and verify that both sides of the equation are equal:

Original equation: $3(-y+7)=3(y+5)+6$

Substitute $y = 0$:

3(−0+7)=3(0+5)+63(-0+7)=3(0+5)+6

Simplify:

3(7)=3(5)+63(7)=3(5)+6

21=15+621 = 15 + 6

21=2121 = 21

Since both sides of the equation are equal, our solution $y = 0$ is correct.

Analyzing the Solution

Having solved the equation $3(-y+7)=3(y+5)+6$, we found that the solution is $y = 0$. Now, let's revisit the initial statements and determine which one is true:

A. The equation has one solution, $y=0$. B. The equation has one solution, $y=-1$. C. The equation has no solution. D. The equation has infinitely many solutions.

Based on our step-by-step solution and verification, it is clear that option A is the correct statement. The equation has one solution, and that solution is $y = 0$.

Common Mistakes and How to Avoid Them

When solving equations like this, it's easy to make mistakes if you're not careful. Here are some common mistakes and tips on how to avoid them:

  1. Incorrect Distribution: A common mistake is not distributing the constant correctly across all terms inside the parentheses. For example, in the original equation, one might forget to multiply the 3 by both -y and +7, or by both y and +5. To avoid this, always double-check that you've multiplied the constant by each term inside the parentheses.

  2. Sign Errors: Sign errors are also common, especially when dealing with negative numbers. For example, when distributing $3(-y+7)$, it's crucial to remember that $3 * -y = -3y$. Similarly, when moving terms across the equals sign, remember to change the sign. For example, if you have $3y$ on the right side and want to move it to the left, you need to subtract $3y$ from both sides.

  3. Combining Unlike Terms: Another mistake is trying to combine terms that are not like terms. For example, you can't combine $3y$ and 21 because they are not like terms. Make sure you only combine terms that have the same variable and exponent, or that are constants.

  4. Forgetting to Verify: It's always a good idea to verify your solution by plugging it back into the original equation. This can help you catch any mistakes you might have made along the way. If the solution doesn't make the equation true, you know you need to go back and check your work.

Alternative Approaches to Solving the Equation

While we have demonstrated a detailed step-by-step approach to solving the equation, it's worth noting that there might be alternative methods to reach the same solution. Different approaches can provide a deeper understanding of algebraic principles and enhance problem-solving skills.

Method 1: Simplifying Before Distributing

In some cases, it's possible to simplify the equation before distributing constants. However, in this particular equation, simplification before distribution isn't straightforward due to the structure of the equation. The primary operation required is distribution to eliminate the parentheses and combine like terms.

Method 2: Using Properties of Equality

Throughout the solution, we've applied properties of equality, such as the addition and subtraction properties, which allow us to add or subtract the same value from both sides of the equation without changing its balance. Another property is the division property of equality, which we used to divide both sides by 6 to isolate y. Understanding and applying these properties is crucial in solving any algebraic equation.

Method 3: Recognizing Patterns

As you gain experience in solving equations, you may start recognizing patterns that can help you simplify the process. For example, in this equation, after distributing and simplifying, we reached the equation $21 = 6y + 21$. At this point, one might recognize that subtracting 21 from both sides will directly lead to $0 = 6y$, thus simplifying the steps needed to isolate y. Recognizing such patterns can make the problem-solving process more efficient.

Real-World Applications of Solving Equations

Solving algebraic equations is not just a theoretical exercise; it has numerous practical applications in real life. Understanding how to manipulate equations can help in various fields, from personal finance to engineering. Here are a few examples:

  1. Personal Finance: Equations are used to calculate interest on loans, plan budgets, and determine savings goals. For example, if you want to save a certain amount of money in a specific timeframe, you can use an equation to figure out how much you need to save each month.

  2. Engineering: Engineers use equations to design structures, calculate loads, and analyze systems. Whether it's designing a bridge, a building, or an electrical circuit, equations are at the heart of the process.

  3. Physics: Physics relies heavily on equations to describe the behavior of the physical world. From calculating the trajectory of a projectile to understanding the laws of thermodynamics, equations are essential tools.

  4. Computer Science: Equations are used in algorithms, data analysis, and machine learning. For example, machine learning models use equations to make predictions and classifications based on data.

  5. Chemistry: Chemists use equations to balance chemical reactions, calculate reaction rates, and determine the composition of compounds. Equations help in understanding and predicting chemical behavior.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of solving the equation $3(-y+7)=3(y+5)+6$. By following a step-by-step approach, we distributed constants, combined like terms, and isolated the variable y. We determined that the equation has one solution, $y = 0$, which corresponds to option A. Furthermore, we discussed common mistakes to avoid, alternative solution methods, and real-world applications of solving equations.

Understanding the principles of algebraic manipulation and equation-solving is a fundamental skill that extends beyond the classroom. It empowers individuals to tackle a wide range of problems in various domains, making it a valuable asset in both academic and professional pursuits. By mastering these skills, learners can approach mathematical challenges with confidence and precision.