Solve The Equation: ${ 2(x-2)-(x-1)=2x-2 }$

by ADMIN 45 views

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, 2(x-2)-(x-1)=2x-2, and provide a step-by-step guide on how to simplify it.

Understanding the Equation

The given equation is 2(x-2)-(x-1)=2x-2. To solve this equation, we need to follow the order of operations (PEMDAS) and simplify the expression step by step.

Step 1: Distribute the Coefficients

The first step in solving the equation is to distribute the coefficients to the terms inside the parentheses. In this case, we have 2(x-2) and -(x-1). To distribute the coefficients, we multiply each term inside the parentheses by the coefficient.

2(x-2) = 2x - 4
-(x-1) = -x + 1

Step 2: Combine Like Terms

Now that we have distributed the coefficients, we can combine like terms. In this case, we have 2x - 4 and -x + 1. To combine like terms, we add or subtract the coefficients of the same variable.

2x - 4 - x + 1 = x - 3

Step 3: Simplify the Equation

Now that we have combined like terms, we can simplify the equation by equating the two expressions.

x - 3 = 2x - 2

Step 4: Isolate the Variable

To isolate the variable x, we need to get all the terms with x on one side of the equation. In this case, we can subtract 2x from both sides of the equation.

x - 3 - 2x = -2
-x - 3 = -2

Step 5: Solve for x

Now that we have isolated the variable x, we can solve for x by adding 3 to both sides of the equation.

-x - 3 + 3 = -2 + 3
-x = 1

Step 6: Multiply by -1

To solve for x, we need to multiply both sides of the equation by -1.

-x \* -1 = 1 \* -1
x = -1

Conclusion

In this article, we have solved the linear equation 2(x-2)-(x-1)=2x-2 using the order of operations and simplifying the expression step by step. We have distributed the coefficients, combined like terms, simplified the equation, isolated the variable, and solved for x. The final solution is x = -1.

Frequently Asked Questions

  • What is a linear equation? A linear equation is an equation in which the highest power of the variable is 1.
  • How do I solve a linear equation? To solve a linear equation, you need to follow the order of operations (PEMDAS) and simplify the expression step by step.
  • What is the order of operations (PEMDAS)? The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Tips and Tricks

  • Always follow the order of operations (PEMDAS) when solving linear equations.
  • Simplify the expression step by step to avoid making mistakes.
  • Use the distributive property to distribute coefficients to terms inside parentheses.
  • Combine like terms to simplify the expression.
  • Isolate the variable to solve for x.

Further Reading

  • Linear Equations: A Comprehensive Guide
  • Solving Linear Equations: A Step-by-Step Guide
  • The Order of Operations (PEMDAS): A Guide to Simplifying Expressions

References

  • [1] "Linear Equations" by Math Open Reference
  • [2] "Solving Linear Equations" by Khan Academy
  • [3] "The Order of Operations (PEMDAS)" by Mathway

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 provide a Q&A guide to help you understand and solve linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. It 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 follow the order of operations (PEMDAS) and simplify the expression step by step. This involves distributing coefficients to terms inside parentheses, combining like terms, and isolating the variable.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I distribute coefficients to terms inside parentheses?

A: To distribute coefficients to terms inside parentheses, you multiply each term inside the parentheses by the coefficient. For example, if you have 2(x-2), you would multiply 2 by each term inside the parentheses: 2x - 4.

Q: How do I combine like terms?

A: To combine like terms, you add or subtract the coefficients of the same variable. For example, if you have 2x + 3x, you would combine the like terms by adding the coefficients: 5x.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get all the terms with the variable on one side of the equation. This involves adding or subtracting the same value to both sides of 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.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy both equations. This involves using methods such as substitution or elimination to solve for the variables.

Q: What is the importance of solving linear equations?

A: Solving linear equations is an essential skill in mathematics and is used in a wide range of applications, including science, engineering, economics, and finance.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve linear equations.

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 distributing coefficients to terms inside parentheses
  • Not combining like terms
  • Not isolating the variable
  • Not checking the solution for validity

Q: How can I check my solution for validity?

A: To check your solution for validity, you can plug the values back into the original equation and verify that it is true.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve linear equations. We have covered topics such as the order of operations (PEMDAS), distributing coefficients, combining like terms, isolating the variable, and solving systems of linear equations. We hope that this guide has been helpful in your understanding of linear equations.

Frequently Asked Questions

  • What is a linear equation?
  • How do I solve a linear equation?
  • What is the order of operations (PEMDAS)?
  • How do I distribute coefficients to terms inside parentheses?
  • How do I combine like terms?
  • How do I isolate the variable?
  • What is the difference between a linear equation and a quadratic equation?
  • How do I solve a system of linear equations?
  • What is the importance of solving linear equations?
  • How can I practice solving linear equations?
  • What are some common mistakes to avoid when solving linear equations?
  • How can I check my solution for validity?

Tips and Tricks

  • Always follow the order of operations (PEMDAS) when solving linear equations.
  • Simplify the expression step by step to avoid making mistakes.
  • Use the distributive property to distribute coefficients to terms inside parentheses.
  • Combine like terms to simplify the expression.
  • Isolate the variable to solve for x.
  • Check your solution for validity by plugging the values back into the original equation.

Further Reading

  • Linear Equations: A Comprehensive Guide
  • Solving Linear Equations: A Step-by-Step Guide
  • The Order of Operations (PEMDAS): A Guide to Simplifying Expressions
  • Systems of Linear Equations: A Guide to Solving
  • Linear Algebra: A Guide to Solving Linear Equations and Matrices

References

  • [1] "Linear Equations" by Math Open Reference
  • [2] "Solving Linear Equations" by Khan Academy
  • [3] "The Order of Operations (PEMDAS)" by Mathway
  • [4] "Systems of Linear Equations" by MIT OpenCourseWare
  • [5] "Linear Algebra" by Coursera