Solve The Equation For { X $} : : : { 1 - \frac{x}{6} = \frac{x}{2} - 3 \}

Mar 13, 2025 by ADMIN 75 views

Solving the Equation for x: A Step-by-Step Guide

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Introduction

In this article, we will delve into the world of algebra and solve a linear equation for the variable x. The given equation is 1 - x/6 = x/2 - 3, and our goal is to isolate the variable x and find its value. We will use various algebraic techniques, such as adding, subtracting, multiplying, and dividing, to simplify the equation and solve for x.

Understanding the Equation

The given equation is a linear equation, which means it is an equation in which the highest power of the variable x is 1. The equation is 1 - x/6 = x/2 - 3, and we need to solve for x. To do this, we will use the following steps:

Add or subtract the same value to both sides of the equation to isolate the variable x.
Multiply or divide both sides of the equation by the same non-zero value to eliminate fractions and simplify the equation.
Use inverse operations to isolate the variable x.

Step 1: Add x/6 to Both Sides of the Equation

To isolate the variable x, we need to get rid of the fraction x/6 on the left-hand side of the equation. We can do this by adding x/6 to both sides of the equation. This will give us:

1 - x/6 + x/6 = x/2 - 3 + x/6

Simplifying the left-hand side of the equation, we get:

1 = x/2 - 3 + x/6

Step 2: Add 3 to Both Sides of the Equation

Next, we need to get rid of the constant term -3 on the right-hand side of the equation. We can do this by adding 3 to both sides of the equation. This will give us:

1 + 3 = x/2 - 3 + 3 + x/6

Simplifying the left-hand side of the equation, we get:

4 = x/2 + x/6

Step 3: Multiply Both Sides of the Equation by 6

To eliminate the fractions on the right-hand side of the equation, we can multiply both sides of the equation by 6. This will give us:

6(4) = 6(x/2 + x/6)

Simplifying the left-hand side of the equation, we get:

24 = 3x + x

Step 4: Combine Like Terms on the Right-Hand Side of the Equation

Next, we need to combine the like terms on the right-hand side of the equation. We can do this by adding the x terms together. This will give us:

24 = 4x

Step 5: Divide Both Sides of the Equation by 4

Finally, we need to isolate the variable x by dividing both sides of the equation by 4. This will give us:

24/4 = 4x/4

Simplifying the left-hand side of the equation, we get:

6 = x

Conclusion

In this article, we solved the linear equation 1 - x/6 = x/2 - 3 for the variable x. We used various algebraic techniques, such as adding, subtracting, multiplying, and dividing, to simplify the equation and isolate the variable x. The final solution is x = 6.

Tips and Tricks

When solving linear equations, it is essential to use inverse operations to isolate the variable x.
To eliminate fractions, multiply both sides of the equation by the least common multiple of the denominators.
When combining like terms, add or subtract the coefficients of the like terms.

Real-World Applications

Linear equations have numerous real-world applications in various fields, such as:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future trends.

Final Thoughts

Solving linear equations is a fundamental skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can solve linear equations and isolate the variable x. Remember to use inverse operations, eliminate fractions, and combine like terms to simplify the equation and find the solution.

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Introduction

In this article, we will address some of the most frequently asked questions about solving linear equations. Whether you are a student, a teacher, or simply someone who wants to learn more about linear equations, this article will provide you with the answers you need.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x. You can do this by using inverse operations, such as adding, subtracting, multiplying, and dividing, to simplify the equation and get rid of the constants.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

Not following the order of operations (PEMDAS)
Not using inverse operations to isolate the variable x
Not eliminating fractions by multiplying both sides of the equation by the least common multiple of the denominators
Not combining like terms to simplify the equation

Q: How do I eliminate fractions when solving linear equations?

A: To eliminate fractions when solving linear equations, you need to multiply both sides of the equation by the least common multiple of the denominators. This will get rid of the fractions and make it easier to solve the equation.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1, while a quadratic equation is an equation in which the highest power of the variable is 2. In other words, a linear equation can be written in the form ax + b = c, while a quadratic equation can be written in the form ax^2 + bx + c = 0.

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it is always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: How do I check my work when solving linear equations?

A: To check your work when solving linear equations, you need to plug your solution back into the original equation and make sure that it is true. If it is true, then you have solved the equation correctly.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications in various fields, such as:

Physics: Linear equations are used to describe the motion of objects under constant acceleration.
Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Linear equations are used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving linear equations. Whether you are a student, a teacher, or simply someone who wants to learn more about linear equations, this article will provide you with the answers you need. Remember to use inverse operations, eliminate fractions, and combine like terms to simplify the equation and find the solution.

Tips and Tricks

When solving linear equations, it is essential to use inverse operations to isolate the variable x.
To eliminate fractions, multiply both sides of the equation by the least common multiple of the denominators.
When combining like terms, add or subtract the coefficients of the like terms.