What Is The Solution For The Equation? A + 8 3 = 2 3 A+\frac{8}{3}=\frac{2}{3} A + 3 8 = 3 2 A. A = − 10 3 A=-\frac{10}{3} A = − 3 10 B. A = − 2 A=-2 A = − 2 C. A = 2 A=2 A = 2 D. A = 10 3 A=\frac{10}{3} A = 3 10

Mar 11, 2025 by ADMIN 221 views

**Solving Linear Equations: A Step-by-Step Guide**

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, $a+\frac{8}{3}=\frac{2}{3}$ , and explore the different methods and techniques used to find the solution.

Understanding the Equation

The given equation is a linear equation in one variable, $a$ . The equation is in the form of $a + b = c$ , where $a$ is the variable, $b$ is the constant term, and $c$ is the constant term on the other side of the equation. In this case, $b = \frac{8}{3}$ and $c = \frac{2}{3}$ .

Step 1: Isolate the Variable

To solve the equation, we need to isolate the variable $a$ . This can be done by subtracting the constant term $b$ from both sides of the equation. In this case, we subtract $\frac{8}{3}$ from both sides of the equation.

a + \frac{8}{3} = \frac{2}{3}
a + \frac{8}{3} - \frac{8}{3} = \frac{2}{3} - \frac{8}{3}
a = -\frac{6}{3}

Step 2: Simplify the Expression

Now that we have isolated the variable $a$ , we can simplify the expression by dividing both sides of the equation by the coefficient of $a$ . In this case, the coefficient of $a$ is $1$ , so we can simply divide both sides of the equation by $1$ .

a = -\frac{6}{3}
a = -2

Conclusion

In conclusion, the solution to the equation $a+\frac{8}{3}=\frac{2}{3}$ is $a = -2$ . This can be verified by plugging the value of $a$ back into the original equation.

Answer

The correct answer is:

A. $a=-\frac{10}{3}$ : This is incorrect, as we have already simplified the expression to $a = -2$ .
B. $a=-2$ : This is the correct answer.
C. $a=2$ : This is incorrect, as we have already simplified the expression to $a = -2$ .
D. $a=\frac{10}{3}$ : This is incorrect, as we have already simplified the expression to $a = -2$ .

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS):

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

By following these steps and using the correct order of operations, you can solve linear equations with ease.

Common Mistakes

When solving linear equations, it's easy to make mistakes. Here are some common mistakes to avoid:

Not following the order of operations: Make sure to follow the order of operations (PEMDAS) to avoid errors.
Not isolating the variable: Make sure to isolate the variable on one side of the equation.
Not simplifying the expression: Make sure to simplify the expression by dividing both sides of the equation by the coefficient of the variable.

By avoiding these common mistakes, you can ensure that your solutions are accurate and correct.

Real-World Applications

Linear equations have numerous real-world applications. Here are a few examples:

Physics: Linear equations are used to describe the motion of objects in physics.
Engineering: Linear equations are used to design and optimize systems in engineering.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.

By understanding how to solve linear equations, you can apply this knowledge to real-world problems and make informed decisions.

Conclusion

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 is typically written in the form of 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 isolate the variable on one side of the equation. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

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

Q: How do I isolate the variable?

A: To isolate the variable, you need to get the variable on one side of the equation by itself. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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: 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: How do I check my work?

A: To check your work, you need to plug the solution back into the original equation and verify that it is true. This will help you ensure that your 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:

Not following the order of operations
Not isolating the variable
Not simplifying the expression
Not checking your work

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be applied to a wide range of real-world problems, including:

Physics: Linear equations are used to describe the motion of objects in physics.
Engineering: Linear equations are used to design and optimize systems in engineering.
Economics: Linear equations are used to model economic systems and make predictions about economic trends.

Conclusion

In conclusion, solving linear equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in solving linear equations. Remember to follow the order of operations, isolate the variable, and simplify the expression. By applying this knowledge to real-world problems, you can make informed decisions and solve complex problems.

Additional Resources

For more information on solving linear equations, check out the following resources:

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

Practice Problems

Try solving the following linear equations:

2x + 5 = 11
x - 3 = 7
4x + 2 = 14

Answer Key

x = 3
x = 10
x = 3