Solve The Equation.$\[\frac{x}{3} + 2 = \frac{7}{3}\\]\[$x =\$\]

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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 type of linear equation, which is a simple equation with one variable. We will use the equation x3+2=73\frac{x}{3} + 2 = \frac{7}{3} as an example to demonstrate the step-by-step process of solving a linear equation.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. It can be written in the form ax+b=cax + b = c, where aa, bb, and cc are constants. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to be Solved

The equation we will be solving is x3+2=73\frac{x}{3} + 2 = \frac{7}{3}. This equation is a simple linear equation with one variable, x.

Step 1: Subtract 2 from Both Sides

To solve the equation, we need to isolate the variable x. The first step is to subtract 2 from both sides of the equation. This will eliminate the constant term on the left-hand side of the equation.

x3+2−2=73−2\frac{x}{3} + 2 - 2 = \frac{7}{3} - 2

Simplifying the equation, we get:

x3=73−2\frac{x}{3} = \frac{7}{3} - 2

Step 2: Simplify the Right-Hand Side

To simplify the right-hand side of the equation, we need to find a common denominator for the fractions. In this case, the common denominator is 3.

73−2=73−63\frac{7}{3} - 2 = \frac{7}{3} - \frac{6}{3}

Simplifying further, we get:

x3=13\frac{x}{3} = \frac{1}{3}

Step 3: Multiply Both Sides by 3

To isolate the variable x, we need to multiply both sides of the equation by 3. This will eliminate the fraction on the left-hand side of the equation.

3×x3=3×133 \times \frac{x}{3} = 3 \times \frac{1}{3}

Simplifying the equation, we get:

x=1x = 1

Conclusion

In this article, we solved the linear equation x3+2=73\frac{x}{3} + 2 = \frac{7}{3} using the step-by-step process of subtracting 2 from both sides, simplifying the right-hand side, and multiplying both sides by 3. The final solution is x = 1.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is simplified correctly.
  • When subtracting or adding fractions, it's crucial to find a common denominator to avoid errors.
  • When multiplying or dividing fractions, it's essential to multiply or divide the numerators and denominators separately.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • 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 that is essential for students to master. By following the step-by-step process outlined in this article, students can solve linear equations with confidence. Remember to always follow the order of operations, find a common denominator when subtracting or adding fractions, and multiply or divide fractions separately. With practice and patience, students can become proficient in solving linear equations and apply their skills to real-world problems.

Frequently Asked Questions

Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x by following the step-by-step process of subtracting or adding constants, simplifying the right-hand side, and multiplying or dividing fractions.

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

A: Linear equations have numerous real-world applications, including physics, engineering, and economics.

Q: Why is it essential to follow the order of operations when solving linear equations?

A: Following the order of operations (PEMDAS) ensures that the equation is simplified correctly and avoids errors when subtracting or adding fractions.

Q: How do I find a common denominator when subtracting or adding fractions?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the denominators and rewrite the fractions with the common denominator.

Q: Why is it essential to multiply or divide fractions separately?

A: Multiplying or dividing fractions separately ensures that the equation is simplified correctly and avoids errors when multiplying or dividing fractions.

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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 address some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

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

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x by following the step-by-step process of subtracting or adding constants, simplifying the right-hand side, and multiplying or dividing fractions.

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

A: Linear equations have numerous real-world applications, including physics, engineering, and economics.

Q: Why is it essential to follow the order of operations when solving linear equations?

A: Following the order of operations (PEMDAS) ensures that the equation is simplified correctly and avoids errors when subtracting or adding fractions.

Q: How do I find a common denominator when subtracting or adding fractions?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the denominators and rewrite the fractions with the common denominator.

Q: Why is it essential to multiply or divide fractions separately?

A: Multiplying or dividing fractions separately ensures that the equation is simplified correctly and avoids errors when multiplying or dividing fractions.

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 (in this case, x) 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 quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where aa, bb, and cc are constants.

Q: What is the significance of the variable x in a linear equation?

A: The variable x represents the unknown value that we are trying to solve for in a linear equation.

Q: How do I check my solution to a linear equation?

A: To check your solution, you need to plug the value of x back into the original equation and verify that it is true.

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
  • Not finding a common denominator when subtracting or adding fractions
  • Not multiplying or dividing fractions separately
  • Not checking the solution to the equation

Q: How do I practice solving linear equations?

A: To practice solving linear equations, you can try solving a variety of problems, including simple equations and more complex equations. You can also use online resources, such as worksheets and practice tests, to help you improve your skills.

Q: What are some resources available for learning more about solving linear equations?

A: Some resources available for learning more about solving linear equations include:

  • Online tutorials and videos
  • Textbooks and workbooks
  • Online practice tests and quizzes
  • Math apps and software

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 step-by-step process outlined in this article, you can solve linear equations with confidence. Remember to always follow the order of operations, find a common denominator when subtracting or adding fractions, and multiply or divide fractions separately. With practice and patience, you can become proficient in solving linear equations and apply your skills to real-world problems.

Final Thoughts

Solving linear equations is a crucial skill that is essential for students to master. By understanding the steps involved in solving these equations, students can apply their skills to a variety of real-world problems. Whether you are a student or a professional, solving linear equations is an essential skill that can help you succeed in your career and personal life.

Additional Resources

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram Alpha: Linear Equations
  • MIT OpenCourseWare: Linear Algebra

Glossary

  • Linear Equation: An equation in which the highest power of the variable (in this case, x) is 1.
  • Quadratic Equation: An equation in which the highest power of the variable is 2.
  • Variable: A symbol or letter that represents a value or quantity.
  • Constant: A value or quantity that does not change.
  • Coefficient: A number or value that is multiplied by a variable.
  • Term: A single part of an expression or equation.
  • Expression: A group of terms that are combined using mathematical operations.
  • Equation: A statement that two expressions are equal.