Solve The Equation:$\frac{2}{3} X = -\frac{1}{2} X + 5$1. What Is The First Step To Isolate The Variable Term On One Side Of The Equation? $\square$2. What Is The Second Step To Solve For $x$? $\square$

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Introduction

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 a specific linear equation, 23x=−12x+5\frac{2}{3} x = -\frac{1}{2} x + 5, and provide a step-by-step guide on how to isolate the variable term on one side of the equation.

Step 1: Add or Subtract the Same Value to Both Sides

The first step to isolate the variable term on one side of the equation is to add or subtract the same value to both sides. In this case, we need to get rid of the term −12x-\frac{1}{2} x on the right-hand side. To do this, we can add 12x\frac{1}{2} x to both sides of the equation.

23x+12x=−12x+12x+5\frac{2}{3} x + \frac{1}{2} x = -\frac{1}{2} x + \frac{1}{2} x + 5

This simplifies to:

23x+12x=5\frac{2}{3} x + \frac{1}{2} x = 5

Step 2: Combine Like Terms

The next step is to combine like terms on the left-hand side of the equation. In this case, we have two terms with the variable xx, 23x\frac{2}{3} x and 12x\frac{1}{2} x. To combine these terms, we need to find a common denominator, which is 6.

46x+36x=5\frac{4}{6} x + \frac{3}{6} x = 5

This simplifies to:

76x=5\frac{7}{6} x = 5

Step 3: Multiply Both Sides by the Reciprocal of the Coefficient

The final step is to multiply both sides of the equation by the reciprocal of the coefficient of xx. In this case, the coefficient of xx is 76\frac{7}{6}, so we need to multiply both sides by 67\frac{6}{7}.

67×76x=67×5\frac{6}{7} \times \frac{7}{6} x = \frac{6}{7} \times 5

This simplifies to:

x=307x = \frac{30}{7}

Conclusion

Solving linear equations requires a step-by-step approach, and in this article, we have provided a guide on how to isolate the variable term on one side of the equation. By following these steps, we can solve equations like 23x=−12x+5\frac{2}{3} x = -\frac{1}{2} x + 5 and find the value of the variable xx.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) and simplify the equation step by step.
  • When combining like terms, make sure to find a common denominator to avoid errors.
  • When multiplying both sides by the reciprocal of the coefficient, make sure to multiply both sides by the same value to avoid errors.

Common Mistakes

  • Failing to simplify the equation step by step can lead to errors and incorrect solutions.
  • Failing to find a common denominator when combining like terms can lead to errors and incorrect solutions.
  • Failing to multiply both sides by the same value when multiplying by the reciprocal of the coefficient can lead to errors and incorrect solutions.

Real-World Applications

Solving linear equations has numerous real-world applications, including:

  • Physics: Solving linear equations is essential in physics to describe the motion of objects and calculate forces and energies.
  • Engineering: Solving linear equations is essential in engineering to design and optimize systems and structures.
  • Economics: Solving linear equations is essential in economics to model and analyze economic systems and make predictions about future trends.

Conclusion

Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. In this article, we will provide a Q&A guide on solving linear equations, covering common questions and topics related to this subject.

Q: What is a linear equation?

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

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Add or subtract the same value to both sides to isolate the variable term.
  2. Combine like terms on the left-hand side of the equation.
  3. Multiply both sides by the reciprocal of the coefficient of the variable.

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: To simplify an equation, follow these steps:

  1. Combine like terms on the left-hand side of the equation.
  2. Simplify any fractions or decimals.
  3. Check if the equation can be simplified further.

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. For example, the equation x + 2 = 3 is a linear equation, while the equation x^2 + 4x + 4 = 0 is a quadratic equation.

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

A: Yes, you can use a calculator to solve linear equations. However, it's essential to understand the steps involved in solving the equation and to check your work to ensure that the solution is correct.

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

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

  • Failing to simplify the equation step by step.
  • Failing to find a common denominator when combining like terms.
  • Failing to multiply both sides by the same value when multiplying by the reciprocal of the coefficient.

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

A: To check your work when solving a linear equation, follow these steps:

  1. Plug the solution back into the original equation.
  2. Simplify the equation and check if it is true.
  3. If the equation is not true, recheck your work and try again.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations and apply this skill to a wide range of real-world problems.

Additional Resources

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations

Practice Problems

  1. Solve the equation 2x + 3 = 5.
  2. Solve the equation x - 2 = 3.
  3. Solve the equation 4x + 2 = 10.

Answer Key

  1. x = 1
  2. x = 5
  3. x = 2.25