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Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear equations, specifically the equation (2x-5) + Ix - 20 = -2(-5x + 20). We will break down the steps to solve this equation and provide a clear understanding of the concept.

Understanding the Equation

The given equation is (2x-5) + Ix - 20 = -2(-5x + 20). To solve this equation, we need to simplify it and isolate the variable x. The equation involves addition, subtraction, multiplication, and division operations.

Step 1: Simplify the Equation

To simplify the equation, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses: (2x-5) + Ix - 20 = -2(-5x + 20)
  2. Simplify the expressions: 2x - 5 + ix - 20 = 10x - 40
  3. Combine like terms: (2x + ix) - 25 = 10x - 40

Step 2: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable x. To do this, we need to get all the terms with x on one side of the equation and the constant terms on the other side.

Step 3: Add or Subtract Constants

To isolate the variable x, we need to add or subtract constants to both sides of the equation. In this case, we need to add 25 to both sides of the equation:

(2x + ix) - 25 + 25 = 10x - 40 + 25

Step 4: Simplify the Equation

After adding 25 to both sides of the equation, we get:

(2x + ix) = 10x + 5

Step 5: Combine Like Terms

Now that we have simplified the equation, we can combine like terms:

2x + ix = 10x + 5

Step 6: Isolate the Variable

To isolate the variable x, we need to get all the terms with x on one side of the equation and the constant terms on the other side. We can do this by subtracting 10x from both sides of the equation:

ix - 8x = 5

Step 7: Factor Out the Common Term

Now that we have isolated the variable x, we can factor out the common term x:

x(i - 8) = 5

Step 8: Solve for x

To solve for x, we need to divide both sides of the equation by (i - 8):

x = 5 / (i - 8)

Conclusion

In this article, we have solved the linear equation (2x-5) + Ix - 20 = -2(-5x + 20). We have broken down the steps to solve this equation and provided a clear understanding of the concept. Solving linear equations is a crucial skill for students to master, and with practice, they can become proficient in solving these types of equations.

Tips and Tricks

  • Always follow the order of operations (PEMDAS) when simplifying equations.
  • Isolate the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  • Use factoring to simplify equations and isolate the variable.
  • Practice solving linear equations to become proficient in this skill.

Real-World Applications

Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects.
  • Engineering: Linear equations are used to design and optimize systems.
  • Economics: Linear equations are used to model economic systems and make predictions.

Common Mistakes

  • Not following the order of operations (PEMDAS) when simplifying equations.
  • Not isolating the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  • Not using factoring to simplify equations and isolate the variable.

Conclusion

Solving linear equations is a crucial skill for students to master. With practice and patience, they can become proficient in solving these types of equations. Remember to always follow the order of operations (PEMDAS), isolate the variable, and use factoring to simplify equations and isolate the variable.

Introduction

In our previous article, we solved the linear equation (2x-5) + Ix - 20 = -2(-5x + 20). In this article, we will provide a Q&A section to help students understand the concept of solving linear equations better.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to follow the order of operations (PEMDAS):

  1. Evaluate the expressions inside the parentheses.
  2. Simplify the expressions.
  3. Combine like terms.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to get all the terms with the variable on one side of the equation and the constant terms on the other side. You can do this by adding or subtracting constants to both sides of the equation.

Q: What is factoring in linear equations?

A: Factoring in linear equations is the process of expressing an equation as a product of two or more factors. It is used to simplify equations and isolate the variable.

Q: How do I use factoring to simplify a linear equation?

A: To use factoring to simplify a linear equation, you need to identify the common factors in the equation and express it as a product of two or more factors.

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) when simplifying equations.
  • Not isolating the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  • Not using factoring to simplify equations and isolate the variable.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Working on sample problems and exercises.
  • Using online resources and tools to practice solving linear equations.
  • Joining a study group or working with a tutor to practice solving linear equations.

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

A: Linear equations have many real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects.
  • Engineering: Linear equations are used to design and optimize systems.
  • Economics: Linear equations are used to model economic systems and make predictions.

Conclusion

Solving linear equations is a crucial skill for students to master. With practice and patience, they can become proficient in solving these types of equations. Remember to always follow the order of operations (PEMDAS), isolate the variable, and use factoring to simplify equations and isolate the variable.

Tips and Tricks

  • Always follow the order of operations (PEMDAS) when simplifying equations.
  • Isolate the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  • Use factoring to simplify equations and isolate the variable.
  • Practice solving linear equations to become proficient in this skill.

Common Mistakes

  • Not following the order of operations (PEMDAS) when simplifying equations.
  • Not isolating the variable by getting all the terms with the variable on one side of the equation and the constant terms on the other side.
  • Not using factoring to simplify equations and isolate the variable.

Conclusion

Solving linear equations is a crucial skill for students to master. With practice and patience, they can become proficient in solving these types of equations. Remember to always follow the order of operations (PEMDAS), isolate the variable, and use factoring to simplify equations and isolate the variable.