Solve For { A $} : : : { 11 = \frac{1}{3} A + 2 \}

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Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate variables in equations. In this article, we will focus on solving a linear equation that involves fractions. The given equation is ${ 11 = \frac{1}{3} a + 2 }$, and our goal is to solve for the variable ${ a }$.

Understanding the Equation

Before we start solving the equation, let's break it down and understand what it means. The equation states that ${ 11 }$ is equal to the sum of ${ \frac{1}{3} }$ times ${ a }$ and ${ 2 }$. In other words, we have a fraction multiplied by ${ a }$ and added to a constant value, which is ${ 2 }$.

Isolating the Variable

To solve for ${ a }$, we need to isolate the variable on one side of the equation. We can start by subtracting ${ 2 }$ from both sides of the equation. This will help us get rid of the constant term on the right-hand side.

Subtracting 2 from Both Sides

${ 11 - 2 = \frac{1}{3} a + 2 - 2 }$

${ 9 = \frac{1}{3} a }$

Multiplying Both Sides by 3

To get rid of the fraction, we can multiply both sides of the equation by ${ 3 }$. This will help us isolate the variable ${ a }$.

${ 3 \times 9 = 3 \times \frac{1}{3} a }$

${ 27 = a }$

Conclusion

In this article, we solved a linear equation that involved fractions. We started by understanding the equation and then isolated the variable by subtracting ${ 2 }$ from both sides and multiplying both sides by ${ 3 }$. The final solution is ${ a = 27 }$.

Tips and Tricks

• When solving linear equations, it's essential to follow the order of operations (PEMDAS).
• Make sure to isolate the variable on one side of the equation.
• Use fractions and decimals to simplify the equation.
• Check your solution by plugging it back into the original equation.

Real-World Applications

Solving linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. Here are a few examples:

• Finance: Linear equations are used to calculate interest rates, investments, and loans.
• Science: Linear equations are used to model population growth, chemical reactions, and physical systems.
• Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable on one side of the equation can lead to incorrect solutions.
• Not checking the solution: Failing to check the solution by plugging it back into the original equation can lead to incorrect solutions.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can solve linear equations involving fractions and decimals. Remember to follow the order of operations, isolate the variable, and check your solution. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

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Introduction to Q&A

In the previous article, we solved a linear equation that involved fractions. We received many questions from readers who wanted to know more about solving linear equations and how to apply them to real-world problems. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q&A

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation that can be written in the form ${ ax + b = c }$, where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ x }$ is the variable. A quadratic equation, on the other hand, is an equation that can be written in the form ${ ax^2 + bx + c = 0 }$, where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ x }$ is the variable.

Q: How do I know if an equation is linear or quadratic?

A: To determine if an equation is linear or quadratic, look for the highest power of the variable. If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

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

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any 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 solve a linear equation with fractions?

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

1. Multiply both sides of the equation by the least common multiple (LCM) of the denominators.
2. Simplify the equation by canceling out any common factors.
3. Isolate the variable by adding or subtracting the same value to both sides of the equation.
4. Check your solution by plugging it back into the original equation.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is an equation that can be written in the form ${ ax + b = c }$, where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ x }$ is the variable. A system of linear equations is a set of two or more linear equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, follow these steps:

1. Write the system of equations in the form ${ ax + by = c }$.
2. Use the method of substitution or elimination to solve for one variable.
3. Substitute the value of the variable into one of the original equations to solve for the other variable.
4. Check your solution by plugging it back into the original equations.

Conclusion

In this article, we answered some of the most frequently asked questions about solving linear equations. We covered topics such as the difference between linear and quadratic equations, the order of operations (PEMDAS), and how to solve linear equations with fractions. We also discussed the difference between a linear equation and a system of linear equations, and how to solve a system of linear equations. By following the steps outlined in this article, you can become proficient in solving linear equations and apply them to real-world problems.

Tips and Tricks

• Make sure to follow the order of operations (PEMDAS) when solving linear equations.
• Use fractions and decimals to simplify the equation.
• Check your solution by plugging it back into the original equation.
• Use the method of substitution or elimination to solve a system of linear equations.

Real-World Applications

Solving linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. Here are a few examples:

• Finance: Linear equations are used to calculate interest rates, investments, and loans.
• Science: Linear equations are used to model population growth, chemical reactions, and physical systems.
• Engineering: Linear equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

• Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect solutions.
• Not isolating the variable: Failing to isolate the variable on one side of the equation can lead to incorrect solutions.
• Not checking the solution: Failing to check the solution by plugging it back into the original equation can lead to incorrect solutions.

Final Thoughts

Solving linear equations is a fundamental concept in mathematics, and it has numerous real-world applications. By following the steps outlined in this article, you can become proficient in solving linear equations and apply them to real-world problems. Remember to follow the order of operations, isolate the variable, and check your solution. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.