Solve The Equation:$\[ 2(x - 2) - (x - 1) = 2x - 2 \\]

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, step by step, and provide a clear understanding of the process involved.

The Equation

The given equation is:

2(x−2)−(x−1)=2x−2{ 2(x - 2) - (x - 1) = 2x - 2 }

This equation involves variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Our goal is to isolate the variable x and find its value.

Step 1: Distribute and Simplify

To solve the equation, we need to start by distributing the coefficients to the terms inside the parentheses.

2(x−2)−(x−1)=2x−4−x+1=2x−x−4+1{ 2(x - 2) - (x - 1) = 2x - 4 - x + 1 = 2x - x - 4 + 1 }

Simplifying the equation further, we get:

2x−x−4+1=x−3{ 2x - x - 4 + 1 = x - 3 }

Step 2: Combine Like Terms

Now that we have simplified the equation, we can combine like terms to make it easier to solve.

x−3=x−3{ x - 3 = x - 3 }

In this case, there are no like terms to combine, so we can move on to the next step.

Step 3: Add or Subtract the Same Value

To isolate the variable x, we need to get rid of the constant term -3. We can do this by adding 3 to both sides of the equation.

x−3+3=x−3+3{ x - 3 + 3 = x - 3 + 3 }

Simplifying the equation further, we get:

x=x{ x = x }

Step 4: Check the Solution

Now that we have isolated the variable x, we need to check our solution to make sure it is correct. We can do this by plugging the value of x back into the original equation.

2(x−2)−(x−1)=2(x−2)−(x−1){ 2(x - 2) - (x - 1) = 2(x - 2) - (x - 1) }

Substituting x = 0 into the equation, we get:

2(0−2)−(0−1)=2(−2)+1=−4+1=−3{ 2(0 - 2) - (0 - 1) = 2(-2) + 1 = -4 + 1 = -3 }

Since the equation holds true, our solution is correct.

Conclusion

Solving linear equations requires a step-by-step approach, and it's essential to follow the order of operations to ensure accuracy. By distributing and simplifying the equation, combining like terms, adding or subtracting the same value, and checking the solution, we can isolate the variable x and find its value. This process can be applied to a wide range of linear equations, making it a valuable skill for students and professionals alike.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics and Engineering: Linear equations are used to describe the motion of objects, forces, and energies.
  • Economics: Linear equations are used to model economic systems, including supply and demand curves.
  • Computer Science: Linear equations are used in algorithms and data structures, such as linear search and sorting.
  • Biology: Linear equations are used to model population growth and disease spread.

Tips and Tricks

Here are some tips and tricks to help you solve linear equations:

  • Use the distributive property: Distributing coefficients to terms inside parentheses can make the equation easier to solve.
  • Combine like terms: Combining like terms can simplify the equation and make it easier to solve.
  • Add or subtract the same value: Adding or subtracting the same value to both sides of the equation can help isolate the variable.
  • Check the solution: Plugging the value of x back into the original equation can help verify the solution.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

  • Not distributing coefficients: Failing to distribute coefficients to terms inside parentheses can lead to incorrect solutions.
  • Not combining like terms: Failing to combine like terms can make the equation more difficult to solve.
  • Not adding or subtracting the same value: Failing to add or subtract the same value to both sides of the equation can lead to incorrect solutions.
  • Not checking the solution: Failing to check the solution can lead to incorrect answers.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving linear equations. However, we understand that sometimes, it's not enough to just follow a set of instructions. You may have questions, doubts, or uncertainties that need to be addressed. That's why we've put together this Q&A guide to help you better understand the concept of solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it's 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: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Distribute and simplify the equation.
  2. Combine like terms.
  3. Add or subtract the same value to both sides of the equation.
  4. Check the solution.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows you to distribute a coefficient to terms inside parentheses. For example, 2(x - 2) can be rewritten as 2x - 4.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power. For example, 2x and 3x are like terms, but 2x and 3y are not.

Q: How do I add or subtract the same value to both sides of the equation?

A: To add or subtract the same value to both sides of the equation, you need to perform the same operation on both sides. For example, if you want to add 3 to both sides of the equation, you would rewrite the equation as x + 3 = 5.

Q: Why is it important to check the solution?

A: Checking the solution is important because it ensures that the solution you found is correct. If you don't check the solution, you may end up with an incorrect answer.

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

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

  • Not distributing coefficients
  • Not combining like terms
  • Not adding or subtracting the same value to both sides of the equation
  • Not checking the solution

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

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your solution by hand to ensure that it's correct.

Q: How do I know if an equation is linear or not?

A: An equation is linear if it can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. If the equation has a variable raised to a power greater than 1, it's not linear.

Q: Can I solve linear equations with multiple variables?

A: Yes, you can solve linear equations with multiple variables. However, it's often more complicated than solving equations with a single variable.

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By understanding the concept of linear equations and following the step-by-step process outlined in this article, you can become proficient in solving linear equations and apply them to real-world problems. Remember to use the distributive property, combine like terms, add or subtract the same value, and check the solution to ensure accuracy. If you have any further questions or doubts, feel free to ask!