Solving 6x + 1 - 2(-7x - 1) = 2(x - 6) A Step-by-Step Guide

July 11, 2025 by Scholario Team 60 views

Solving for x in the Equation 6x + 1 - 2(-7x - 1) = 2(x - 6)

Introduction

In this comprehensive article, we will walk through the step-by-step process of solving for the variable x in the given algebraic equation: 6x + 1 - 2(-7x - 1) = 2(x - 6). Algebraic equations are fundamental in mathematics, and mastering the techniques to solve them is crucial for various applications in science, engineering, and everyday problem-solving. This detailed guide aims to provide a clear understanding of each step involved, ensuring that you can confidently tackle similar problems in the future. We will begin by simplifying the equation, then isolate the variable x, and finally arrive at the solution. By following this structured approach, you will not only find the value of x but also gain a deeper insight into the principles of algebraic manipulation.

Step-by-Step Solution

1. Distribute the Constants

The first step in solving the equation is to distribute the constants outside the parentheses to the terms inside. This involves multiplying each term within the parentheses by the constant factor preceding it. For our equation, 6x + 1 - 2(-7x - 1) = 2(x - 6), we have two instances where we need to apply the distributive property.

First, we distribute -2 to the terms inside the first parentheses (-7x - 1):

-2 * (-7x) = 14x

-2 * (-1) = 2

So, -2(-7x - 1) becomes 14x + 2.

Next, we distribute 2 to the terms inside the second parentheses (x - 6):

2 * x = 2x

2 * (-6) = -12

Thus, 2(x - 6) becomes 2x - 12.

Now, we can rewrite the original equation with these simplifications:

6x + 1 + 14x + 2 = 2x - 12

This step is crucial as it eliminates the parentheses, making the equation easier to handle and allowing us to combine like terms in the subsequent steps.

2. Combine Like Terms

After distributing the constants, the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (e.g., 6x and 14x) or are constants (e.g., 1 and 2). Combining like terms simplifies the equation further, making it easier to isolate the variable x.

On the left side of the equation, 6x + 1 + 14x + 2, we have the following like terms:

6x and 14x are like terms because they both contain the variable x raised to the first power.
1 and 2 are like terms because they are both constants.

Combining these terms, we get:

6x + 14x = 20x

1 + 2 = 3

So, the left side of the equation simplifies to 20x + 3.

On the right side of the equation, 2x - 12, there are no like terms to combine, as 2x contains the variable x and -12 is a constant. Thus, the right side remains 2x - 12.

Now, our equation looks like this:

20x + 3 = 2x - 12

This simplified form allows us to proceed with isolating the variable x on one side of the equation.

3. Isolate the Variable Term

The next critical step in solving for x is to isolate the variable term on one side of the equation. This means getting all terms containing x on one side and all constants on the other side. To achieve this, we will use inverse operations to move terms across the equals sign. In our equation, 20x + 3 = 2x - 12, we have variable terms on both sides, so we need to choose which side to consolidate them on.

Generally, it is a good practice to move the variable term with the smaller coefficient to the side with the larger coefficient to avoid dealing with negative coefficients. In this case, we have 20x on the left and 2x on the right. Since 2x has a smaller coefficient, we will subtract 2x from both sides of the equation to move it to the left side:

20x + 3 - 2x = 2x - 12 - 2x

Simplifying this, we get:

18x + 3 = -12

Now, we have the variable term 18x on the left side, and we need to move the constant term +3 to the right side. To do this, we subtract 3 from both sides of the equation:

18x + 3 - 3 = -12 - 3

Simplifying, we get:

18x = -15

At this point, we have successfully isolated the variable term 18x on the left side and the constant -15 on the right side. The equation is now in a form where we can easily solve for x.

4. Solve for x

After isolating the variable term, the final step is to solve for x by dividing both sides of the equation by the coefficient of x. In our equation, 18x = -15, the coefficient of x is 18. To isolate x, we will divide both sides by 18:

18x / 18 = -15 / 18

Simplifying the left side, 18x divided by 18 is simply x:

x = -15 / 18

Now, we need to simplify the fraction -15/18. Both 15 and 18 are divisible by 3, so we can reduce the fraction:

-15 / 3 = -5

18 / 3 = 6

Thus, the simplified fraction is -5/6.

Therefore, the solution for x is:

x = -5/6

We have successfully solved for x by performing the necessary algebraic manipulations. This solution can be verified by substituting it back into the original equation to ensure that both sides of the equation are equal.

Verification

To ensure the correctness of our solution, it is essential to verify that the value we found for x, which is x = -5/6, satisfies the original equation. Verification involves substituting this value back into the original equation and checking if both sides of the equation are equal.

Our original equation is:

6x + 1 - 2(-7x - 1) = 2(x - 6)

Substitute x = -5/6 into the equation:

6(-5/6) + 1 - 2(-7(-5/6) - 1) = 2((-5/6) - 6)

Now, we simplify both sides step by step.

Left Side Simplification

First, we simplify the term 6(-5/6):

6 * (-5/6) = -5

So, the left side becomes:

-5 + 1 - 2(-7(-5/6) - 1)

Next, we simplify the term -7(-5/6):

-7 * (-5/6) = 35/6

Now, the left side looks like:

-5 + 1 - 2(35/6 - 1)

We need to subtract 1 from 35/6. To do this, we convert 1 to a fraction with a denominator of 6:

1 = 6/6

So, 35/6 - 1 becomes:

35/6 - 6/6 = 29/6

Now, the left side is:

-5 + 1 - 2(29/6)

Next, we multiply -2 by 29/6:

-2 * (29/6) = -58/6

Simplifying -58/6 by dividing both the numerator and the denominator by 2, we get:

-58/6 = -29/3

Now, the left side is:

-5 + 1 - 29/3

Combine -5 and 1:

-5 + 1 = -4

So, the left side is:

-4 - 29/3

To combine -4 and -29/3, we need to convert -4 to a fraction with a denominator of 3:

-4 = -12/3

Now, we can subtract:

-12/3 - 29/3 = -41/3

Thus, the left side of the equation simplifies to -41/3.

Right Side Simplification

Now, let's simplify the right side of the equation:

2((-5/6) - 6)

First, we need to subtract 6 from -5/6. To do this, we convert 6 to a fraction with a denominator of 6:

6 = 36/6

So, -5/6 - 6 becomes:

-5/6 - 36/6 = -41/6

Now, the right side is:

2(-41/6)

Multiply 2 by -41/6:

2 * (-41/6) = -82/6

Simplify -82/6 by dividing both the numerator and the denominator by 2, we get:

-82/6 = -41/3

Thus, the right side of the equation simplifies to -41/3.

Conclusion of Verification

We have simplified both sides of the equation after substituting x = -5/6:

Left Side: -41/3

Right Side: -41/3

Since both sides are equal, the value x = -5/6 is indeed the correct solution to the equation.

Common Mistakes and How to Avoid Them

Solving algebraic equations involves several steps, and it’s common for students to make mistakes along the way. Identifying these common pitfalls and understanding how to avoid them can significantly improve your accuracy and confidence in solving equations. Here, we discuss some frequent errors made while solving equations like 6x + 1 - 2(-7x - 1) = 2(x - 6) and strategies to prevent them.

1. Incorrect Distribution

Mistake: One of the most common errors is mishandling the distributive property. This usually involves not distributing the constant to all terms inside the parentheses or incorrectly applying the sign.

Example: In the equation 6x + 1 - 2(-7x - 1) = 2(x - 6), a mistake could be forgetting to distribute the -2 to both -7x and -1, or incorrectly multiplying the signs.

How to Avoid: Always ensure that the constant outside the parentheses is multiplied by each term inside. Pay close attention to signs. A negative number multiplied by a negative number yields a positive number, and a negative number multiplied by a positive number yields a negative number. Write out each step explicitly to avoid errors, especially when dealing with negative numbers. For instance, rewrite -2(-7x - 1) as -2 * -7x + (-2) * -1.

2. Combining Non-Like Terms

Mistake: Another frequent error is combining terms that are not like terms. Like terms have the same variable raised to the same power (e.g., 6x and 14x) or are constants (e.g., 1 and 2). Combining unlike terms leads to an incorrect simplification of the equation.

Example: Trying to combine 20x and 3 in the expression 20x + 3 is a common mistake.

How to Avoid: Before combining terms, identify the like terms carefully. Circle or underline like terms to visually group them. Remember, you can only add or subtract terms that have the same variable raised to the same power or are constants. For example, in the expression 6x + 14x + 1 + 2, combine 6x and 14x to get 20x, and combine 1 and 2 to get 3. The simplified expression is 20x + 3.

3. Incorrectly Applying Inverse Operations

Mistake: When isolating the variable, it’s crucial to apply inverse operations correctly. A common mistake is adding instead of subtracting or multiplying instead of dividing (or vice versa) when moving terms across the equals sign.

Example: In the equation 20x + 3 = 2x - 12, if you add 3 to both sides instead of subtracting, you will not isolate the variable term correctly.

How to Avoid: Always perform the opposite operation to move terms across the equals sign. If a term is added, subtract it from both sides; if a term is subtracted, add it to both sides. If a term is multiplied, divide both sides by it; if a term is divided, multiply both sides by it. Write each step clearly to ensure you are applying the correct operation. For example, to move +3 from the left side of the equation 20x + 3 = 2x - 12, subtract 3 from both sides: 20x + 3 - 3 = 2x - 12 - 3, which simplifies to 20x = 2x - 15.

4. Forgetting to Perform Operations on Both Sides

Mistake: One of the fundamental rules of algebra is that whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality. Forgetting to do this is a frequent mistake.

Example: In the equation 18x = -15, if you divide only the left side by 18 but not the right side, you will arrive at an incorrect solution.

How to Avoid: Always perform the same operation on both sides of the equation. When isolating the variable, make sure that every addition, subtraction, multiplication, or division is applied to both sides. This maintains the balance of the equation and leads to the correct solution. For example, to solve 18x = -15, divide both sides by 18: 18x / 18 = -15 / 18, which simplifies to x = -15/18.

5. Sign Errors

Mistake: Sign errors are particularly common, especially when dealing with negative numbers. These errors can occur during distribution, combining like terms, or applying inverse operations.

Example: Incorrectly simplifying -2(-7x - 1) as -14x - 2 instead of +14x + 2 is a sign error.

How to Avoid: Pay meticulous attention to signs at each step. When distributing, remember that a negative times a negative is a positive, and a negative times a positive is a negative. When combining like terms, ensure you are correctly adding or subtracting the coefficients, taking the signs into account. Writing out each step and double-checking the signs can help prevent these errors. For instance, rewrite -2(-7x - 1) as -2 * -7x + (-2) * -1, which equals 14x + 2.

6. Not Simplifying Fractions

Mistake: Failing to simplify fractions in the final answer is another common oversight. The solution should always be expressed in its simplest form.

Example: Leaving the solution as x = -15/18 instead of simplifying it to x = -5/6.

How to Avoid: After finding the solution, check if the fraction can be simplified. Look for common factors in the numerator and the denominator and divide both by their greatest common divisor. This ensures that the answer is in its simplest form. For example, in x = -15/18, both 15 and 18 are divisible by 3, so the fraction simplifies to x = -5/6.

Conclusion

In this article, we have provided a detailed, step-by-step solution for the algebraic equation 6x + 1 - 2(-7x - 1) = 2(x - 6). We began by distributing the constants, then combined like terms, isolated the variable term, and finally solved for x, arriving at the solution x = -5/6. We also emphasized the importance of verifying the solution by substituting it back into the original equation, confirming its correctness.

Furthermore, we addressed common mistakes that students often make when solving algebraic equations and provided practical strategies to avoid them. These include being meticulous with the distributive property, correctly combining like terms, applying inverse operations accurately, performing operations on both sides of the equation, paying close attention to signs, and simplifying fractions.

By understanding and implementing these techniques, you can significantly improve your ability to solve algebraic equations with confidence and accuracy. Consistent practice and attention to detail are key to mastering these skills and succeeding in mathematics and related fields. Remember to review these concepts regularly and apply them to various problems to reinforce your understanding. With dedication and the right approach, you can confidently tackle even the most challenging algebraic problems.