Solve For \[$ X \$\]:$\[ 5x - 2(2x + 6) = 15 \\]

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Introduction

In this article, we will be solving a linear equation involving variables and constants. The equation given is $5x - 2(2x + 6) = 15$. Our goal is to isolate the variable $x$ and find its value. We will use the order of operations and algebraic properties to simplify the equation and solve for $x$.

Understanding the Equation

The given equation is a linear equation, which means it is an equation in which the highest power of the variable is 1. The equation is $5x - 2(2x + 6) = 15$. To solve this equation, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction (PEMDAS).

Step 1: Simplify the Equation

The first step is to simplify the equation by evaluating the expression inside the parentheses. The expression inside the parentheses is $2x + 6$. We can simplify this expression by multiplying $2$ and $x$, and then adding $6$.

# Define the variables
x = sympy.Symbol('x')

# Define the equation
equation = 5*x - 2*(2*x + 6) - 15

# Simplify the equation
simplified_equation = sympy.simplify(equation)

Step 2: Distribute the Negative Sign

The next step is to distribute the negative sign to the terms inside the parentheses. This will give us $-4x - 12$.

# Distribute the negative sign
distributed_equation = -4*x - 12

Step 3: Combine Like Terms

The next step is to combine like terms. We can combine the terms $5x$ and $-4x$ to get $x$. We can also combine the constants $-12$ and $15$ to get $3$.

# Combine like terms
combined_equation = x + 3

Step 4: Solve for $x$

The final step is to solve for $x$. We can do this by isolating $x$ on one side of the equation. Since the equation is already in the form $x = 3$, we can see that the value of $x$ is $3$.

# Solve for x
solution = sympy.solve(combined_equation, x)

Conclusion

In this article, we solved a linear equation involving variables and constants. We used the order of operations and algebraic properties to simplify the equation and solve for $x$. The final solution is $x = 3$.

Example Use Cases

This equation can be used in a variety of real-world applications, such as:

• Finance: In finance, linear equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: In science, linear equations are used to model population growth, chemical reactions, and other physical phenomena.
• Engineering: In engineering, linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

• Use the order of operations: When solving linear equations, it's essential to follow the order of operations (PEMDAS).
• Simplify the equation: Simplifying the equation can make it easier to solve.
• Combine like terms: Combining like terms can help to isolate the variable.
• Check your solution: Always check your solution to make sure it's correct.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

• Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
• Not combining like terms: Failing to combine like terms can make it harder to isolate the variable.
• Not checking your solution: Failing to check your solution can lead to incorrect answers.

Final Thoughts

Solving linear equations is a fundamental skill in mathematics and is used in a variety of real-world applications. By following the order of operations, simplifying the equation, combining like terms, and checking your solution, you can solve linear equations with ease. Remember to always check your solution to make sure it's correct, and don't be afraid to ask for help if you're struggling.

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Introduction

In our previous article, we solved a linear equation involving variables and constants. We used the order of operations and algebraic properties to simplify the equation and solve for $x$. In this article, we will answer some frequently asked questions about solving linear equations.

Q&A

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 we have multiple operations in an expression. The order of operations is:

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 simplify an equation?

A: To simplify an equation, we need to combine like terms and eliminate any unnecessary operations. Here are the steps to simplify an equation:

1. Combine like terms: Combine any terms that have the same variable and coefficient.
2. Eliminate unnecessary operations: Eliminate any operations that are not necessary to solve the equation.
3. Simplify the equation: Simplify the equation by combining like terms and eliminating unnecessary operations.

Q: How do I solve a linear equation?

A: To solve a linear equation, we need to isolate the variable on one side of the equation. Here are the steps to solve a linear equation:

1. Add or subtract the same value to both sides: Add or subtract the same value to both sides of the equation to eliminate any constants.
2. Multiply or divide both sides by the same value: Multiply or divide both sides of the equation by the same value to eliminate any fractions.
3. Simplify the equation: Simplify the equation by combining like terms and eliminating unnecessary operations.

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

A: Here are some common mistakes to avoid when solving linear equations:

• Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
• Not combining like terms: Failing to combine like terms can make it harder to isolate the variable.
• Not checking your solution: Failing to check your solution can lead to incorrect answers.

Q: How do I check my solution?

A: To check your solution, you need to plug the solution back into the original equation and make sure it is true. Here are the steps to check your solution:

1. Plug the solution back into the original equation: Plug the solution back into the original equation.
2. Simplify the equation: Simplify the equation by combining like terms and eliminating unnecessary operations.
3. Check if the equation is true: Check if the equation is true.

Conclusion

Solving linear equations is a fundamental skill in mathematics and is used in a variety of real-world applications. By following the order of operations, simplifying the equation, combining like terms, and checking your solution, you can solve linear equations with ease. Remember to always check your solution to make sure it's correct, and don't be afraid to ask for help if you're struggling.

Example Use Cases

This equation can be used in a variety of real-world applications, such as:

• Finance: In finance, linear equations are used to calculate interest rates, investment returns, and other financial metrics.
• Science: In science, linear equations are used to model population growth, chemical reactions, and other physical phenomena.
• Engineering: In engineering, linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

• Use the order of operations: When solving linear equations, it's essential to follow the order of operations (PEMDAS).
• Simplify the equation: Simplifying the equation can make it easier to solve.
• Combine like terms: Combining like terms can help to isolate the variable.
• Check your solution: Always check your solution to make sure it's correct.

Common Mistakes

Here are some common mistakes to avoid when solving linear equations:

• Not following the order of operations: Failing to follow the order of operations can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it harder to solve.
• Not combining like terms: Failing to combine like terms can make it harder to isolate the variable.
• Not checking your solution: Failing to check your solution can lead to incorrect answers.

Final Thoughts

Solving linear equations is a fundamental skill in mathematics and is used in a variety of real-world applications. By following the order of operations, simplifying the equation, combining like terms, and checking your solution, you can solve linear equations with ease. Remember to always check your solution to make sure it's correct, and don't be afraid to ask for help if you're struggling.