Solve: \[$\frac{2b}{3} = 8\$\]

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Introduction

In mathematics, equations are a fundamental concept that helps us solve problems and understand relationships between variables. Solving equations is a crucial skill that is used in various fields, including physics, engineering, economics, and more. In this article, we will focus on solving a specific equation, {\frac{2b}{3} = 8$}$, and provide a step-by-step guide on how to solve it.

Understanding the Equation

The equation {\frac{2b}{3} = 8$}$ is a linear equation that involves a variable, b{b}, and a constant, 8{8}. The equation is in the form of a fraction, where the numerator is 2b{2b} and the denominator is 3{3}. Our goal is to isolate the variable b{b} and find its value.

Step 1: Multiply Both Sides by 3

To solve the equation, we need to get rid of the fraction. We can do this by multiplying both sides of the equation by the denominator, which is 3{3}. This will eliminate the fraction and give us a simpler equation.

{\frac{2b}{3} = 8$} {\times 3} = {\times 8$} {2b = 24\$}

Step 2: Divide Both Sides by 2

Now that we have eliminated the fraction, we need to isolate the variable b{b}. We can do this by dividing both sides of the equation by the coefficient of b{b}, which is 2{2}. This will give us the value of b{b}.

${2b = 24\$} {\div 2}$ = {\div 24$} {b = 12\$}

Conclusion

In this article, we have solved the equation {\frac{2b}{3} = 8$}$ using a step-by-step approach. We first multiplied both sides of the equation by 3{3} to eliminate the fraction, and then divided both sides by 2{2} to isolate the variable b{b}. The final answer is b=12{b = 12}.

Tips and Tricks

  • When solving equations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • To eliminate fractions, multiply both sides of the equation by the denominator.
  • To isolate a variable, divide both sides of the equation by the coefficient of the variable.

Real-World Applications

Solving equations is a crucial skill that has numerous real-world applications. Here are a few examples:

  • Physics: Solving equations is essential in physics to describe the motion of objects, calculate forces, and understand energy transformations.
  • Engineering: Engineers use equations to design and optimize systems, such as bridges, buildings, and electronic circuits.
  • Economics: Economists use equations to model economic systems, understand the behavior of markets, and make predictions about future trends.

Common Mistakes to Avoid

When solving equations, it's essential to avoid common mistakes that can lead to incorrect solutions. Here are a few examples:

  • Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
  • Not eliminating fractions: Failing to eliminate fractions can make it difficult to solve the equation.
  • Not isolating the variable: Failing to isolate the variable can make it difficult to find its value.

Conclusion

Introduction

In our previous article, we solved the equation {\frac{2b}{3} = 8$}$ using a step-by-step approach. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in solving equations.

Q: What is an equation?

A: An equation is a statement that expresses the equality of two mathematical expressions. It is a fundamental concept in mathematics that helps us solve problems and understand relationships between variables.

Q: What are the different types of equations?

A: There are several types of equations, including:

  • Linear equations: Equations that involve a linear relationship between variables, such as 2x+3=5{2x + 3 = 5}.
  • Quadratic equations: Equations that involve a quadratic relationship between variables, such as x2+4x+4=0{x^2 + 4x + 4 = 0}.
  • Polynomial equations: Equations that involve a polynomial relationship between variables, such as x3+2x2−3x+1=0{x^3 + 2x^2 - 3x + 1 = 0}.

Q: How do I solve a linear equation?

A: To solve a linear equation, follow these steps:

  1. Isolate the variable: Get the variable on one side of the equation by adding, subtracting, multiplying, or dividing both sides by a constant.
  2. Eliminate fractions: Multiply both sides of the equation by the denominator to eliminate fractions.
  3. Simplify the equation: Simplify the equation by combining like terms.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, follow these steps:

  1. Factor the equation: Factor the quadratic expression into the product of two binomials.
  2. Set each factor equal to zero: Set each factor equal to zero and solve for the variable.
  3. Combine the solutions: Combine the solutions to find the final answer.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when evaluating an expression. The order of operations is:

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

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

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

  • Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
  • Not eliminating fractions: Failing to eliminate fractions can make it difficult to solve the equation.
  • Not isolating the variable: Failing to isolate the variable can make it difficult to find its value.

Q: How do I check my solution?

A: To check your solution, plug the value back into the original equation and evaluate it. If the equation is true, then your solution is correct.

Conclusion

Solving equations is a fundamental concept in mathematics that has numerous real-world applications. By following a step-by-step approach and avoiding common mistakes, we can solve equations with confidence. In this article, we have provided a Q&A guide to help you understand the concepts and techniques involved in solving equations.