Solving For Y In -(7/6)y = 14 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article provides a detailed walkthrough of solving for y in the equation -(7/6)y = 14. We will explore the steps involved, offering clear explanations and insights to enhance your understanding. This guide aims to equip you with the knowledge and confidence to tackle similar algebraic problems.

Understanding the Basics of Algebraic Equations

Before diving into the specifics of solving for y, it’s crucial to understand the basics of algebraic equations. An equation is a mathematical statement that asserts the equality of two expressions. It typically involves variables, which are symbols (usually letters) representing unknown values, and constants, which are fixed values. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true.

In our case, the equation is -(7/6)y = 14. Here, y is the variable we need to solve for, and -(7/6) and 14 are constants. The equation states that when y is multiplied by -(7/6), the result is 14. To find y, we need to isolate it on one side of the equation.

Step-by-Step Solution to -(7/6)y = 14

1. Isolate y by Multiplying by the Reciprocal

The key to solving this equation is to isolate y. Since y is being multiplied by a fraction, -(7/6), we can isolate it by multiplying both sides of the equation by the reciprocal of -(7/6). The reciprocal of a fraction is obtained by swapping the numerator and the denominator. The reciprocal of -(7/6) is -(6/7). Remember to maintain the negative sign.

Multiplying both sides by -(6/7) ensures that the equation remains balanced. This is a fundamental principle in algebra: whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain equality.

So, we multiply both sides of the equation -(7/6)y = 14 by -(6/7):

-(6/7) * -(7/6)y = 14 * -(6/7)

2. Simplify the Left Side of the Equation

On the left side of the equation, we have -(6/7) multiplied by -(7/6)y. When we multiply a fraction by its reciprocal, the result is 1. This is because the numerators and denominators cancel each other out. In our case, -(6/7) * -(7/6) equals 1. Therefore, the left side simplifies to 1 * y, which is simply y.

y = 14 * -(6/7)

3. Simplify the Right Side of the Equation

Now, let's simplify the right side of the equation, which is 14 * -(6/7). To multiply a whole number by a fraction, we can treat the whole number as a fraction with a denominator of 1. So, 14 can be written as 14/1.

Now we have (14/1) * -(6/7). To multiply fractions, we multiply the numerators together and the denominators together:

(14/1) * -(6/7) = (14 * -6) / (1 * 7)

= -84 / 7

4. Reduce the Fraction to its Simplest Form

The fraction -84/7 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 84 and 7 is 7. Dividing -84 by 7 gives us -12, and dividing 7 by 7 gives us 1. Therefore, the simplified fraction is -12/1, which is equal to -12.

-84 / 7 = -12

5. State the Solution for y

After simplifying both sides of the equation, we find that:

y = -12

This is our solution. The value of y that satisfies the equation -(7/6)y = 14 is -12.

Verification of the Solution

To ensure the accuracy of our solution, it’s always a good practice to verify it by substituting the value of y back into the original equation. If the equation holds true, then our solution is correct.

Let’s substitute y = -12 into the original equation -(7/6)y = 14:

-(7/6) * -12 = 14

First, we multiply -(7/6) by -12. We can treat -12 as -12/1:

-(7/6) * (-12/1) = (7 * 12) / (6 * 1)

= 84 / 6

Now, we simplify the fraction 84/6 by dividing both the numerator and the denominator by their GCD, which is 6:

84 / 6 = 14

So, the left side of the equation simplifies to 14, which is equal to the right side of the equation. This confirms that our solution y = -12 is correct.

Alternative Methods for Solving the Equation

While multiplying by the reciprocal is a common method, there are other ways to solve the equation -(7/6)y = 14. One alternative method involves multiplying both sides of the equation by the denominator to eliminate the fraction and then dividing by the coefficient of y. Let's explore this method.

1. Multiply Both Sides by the Denominator

In the equation -(7/6)y = 14, the denominator is 6. To eliminate the fraction, we multiply both sides of the equation by 6:

6 * -(7/6)y = 6 * 14

2. Simplify Both Sides

On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with:

-7y = 6 * 14

On the right side, we multiply 6 by 14:

-7y = 84

3. Divide by the Coefficient of y

Now, we have -7y = 84. To isolate y, we divide both sides of the equation by the coefficient of y, which is -7:

-7y / -7 = 84 / -7

4. Simplify to Find y

On the left side, -7y divided by -7 simplifies to y. On the right side, 84 divided by -7 is -12:

y = -12

This method also leads us to the same solution, y = -12, confirming the consistency of algebraic principles.

Common Mistakes to Avoid

When solving equations, it’s crucial to avoid common mistakes that can lead to incorrect answers. Here are a few common pitfalls to watch out for:

1. Incorrectly Applying Operations: Ensure that any operation performed on one side of the equation is also performed on the other side. For example, if you multiply one side by a number, you must multiply the other side by the same number.
2. Sign Errors: Pay close attention to signs, especially when dealing with negative numbers. A simple sign error can completely change the outcome of the equation.
3. Incorrectly Simplifying Fractions: Ensure that fractions are simplified correctly. When multiplying or dividing fractions, make sure to multiply or divide the numerators and denominators correctly.
4. Forgetting to Distribute: If there is a term being multiplied by an expression in parentheses, make sure to distribute the term to each part of the expression.
5. Not Checking the Solution: Always verify your solution by substituting it back into the original equation. This helps catch any errors made during the solving process.

Tips for Mastering Equation Solving

Mastering equation solving requires practice and a solid understanding of algebraic principles. Here are some tips to help you improve your equation-solving skills:

1. Practice Regularly: The more you practice, the more comfortable you will become with solving equations. Try solving a variety of different types of equations to broaden your skills.
2. Understand the Underlying Principles: Don't just memorize steps; understand why you are performing each step. This will help you apply the principles to different types of equations.
3. Break Down Complex Problems: If you encounter a complex equation, break it down into smaller, more manageable steps. This makes the problem less daunting and easier to solve.
4. Show Your Work: Always show your work step-by-step. This helps you keep track of your progress and makes it easier to identify any errors.
5. Check Your Solutions: Always verify your solutions by substituting them back into the original equation. This ensures that your solution is correct.

Conclusion

In this comprehensive guide, we have walked through the process of solving for y in the equation -(7/6)y = 14. We explored the steps involved, from isolating y by multiplying by the reciprocal to simplifying the equation and verifying the solution. We also discussed an alternative method for solving the equation and highlighted common mistakes to avoid. By following the tips and techniques outlined in this guide, you can enhance your equation-solving skills and confidently tackle similar algebraic problems.

Solving equations is a crucial skill in mathematics and has applications in various fields, including science, engineering, and finance. By mastering this skill, you will be well-equipped to solve real-world problems and advance your mathematical knowledge. Keep practicing, and you will become proficient in solving equations of all types.