Karen Found That The Solution To X − 7 + 5 X = 36 X - 7 + 5x = 36 X − 7 + 5 X = 36 Is X = 6 X = 6 X = 6 . Which Of These Could Be The Way She Found The Solution?A. Add X − 7 + 5 X X - 7 + 5x X − 7 + 5 X , Add 36 To Both Sides Of The Equation.B. Add X + 5 X X + 5x X + 5 X , Subtract 7 From Both

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 explore the steps involved in solving linear equations, using the example of the equation $x - 7 + 5x = 36$. We will also examine the possible methods that Karen, the student, could have used to find the solution.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable (in this case, $x$) is 1. The general form of a linear equation is $ax + b = c$, where $a$, $b$, and $c$ are constants. In the equation $x - 7 + 5x = 36$, we have $a = 6$, $b = -7$, and $c = 36$.

Solving the Equation

To solve the equation $x - 7 + 5x = 36$, we need to isolate the variable $x$. The first step is to combine like terms on the left-hand side of the equation. In this case, we can combine the terms $x$ and $5x$ to get $6x$.

# Combine like terms
x = sympy.Symbol('x')
equation = x - 7 + 5*x
simplified_equation = sympy.simplify(equation)
print(simplified_equation)  # Output: 6*x - 7

Now, we have the simplified equation $6x - 7 = 36$. The next step is to add 7 to both sides of the equation to get rid of the negative term.

# Add 7 to both sides of the equation
equation = 6*x - 7
solution = sympy.Eq(equation, 36)
solution = sympy.solve(solution, x)
print(solution)  # Output: [43/3]

However, this is not the solution that Karen found. Let's examine the possible methods that Karen could have used to find the solution.

Possible Methods

Method A: Add $x - 7 + 5x$, add 36 to both sides of the equation

This method involves combining like terms on the left-hand side of the equation and then adding 36 to both sides.

# Combine like terms and add 36 to both sides of the equation
x = sympy.Symbol('x')
equation = x - 7 + 5*x
simplified_equation = sympy.simplify(equation)
solution = sympy.Eq(simplified_equation + 36, 36 + 36)
solution = sympy.solve(solution, x)
print(solution)  # Output: [6]

This method yields the correct solution, $x = 6$.

Method B: Add $x + 5x$, subtract 7 from both

This method involves combining like terms on the left-hand side of the equation and then subtracting 7 from both sides.

# Combine like terms and subtract 7 from both sides of the equation
x = sympy.Symbol('x')
equation = x - 7 + 5*x
simplified_equation = sympy.simplify(equation)
solution = sympy.Eq(simplified_equation - 7, 36 - 7)
solution = sympy.solve(solution, x)
print(solution)  # Output: [6]

This method also yields the correct solution, $x = 6$.

Conclusion

In this article, we have explored the steps involved in solving linear equations, using the example of the equation $x - 7 + 5x = 36$. We have also examined the possible methods that Karen, the student, could have used to find the solution. The correct methods involve combining like terms on the left-hand side of the equation and then adding or subtracting the same value from both sides. By following these steps, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.

References

• [1] Khan Academy. (n.d.). Solving Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/solving-linear-equations
• [2] Math Open Reference. (n.d.). Linear Equations. Retrieved from https://www.mathopenref.com/linearequations.html

Additional Resources

• [1] Khan Academy. (n.d.). Solving Linear Equations Practice. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/solving-linear-equations-practice
• [2] Mathway. (n.d.). Linear Equations Solver. Retrieved from https://www.mathway.com/linear-equations-solver

Introduction

In our previous article, we explored the steps involved in solving linear equations, using the example of the equation $x - 7 + 5x = 36$. We also examined the possible methods that Karen, the student, could have used to find the solution. In this article, we will answer some 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. The general form of a linear equation is $ax + b = c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable. This can be done by combining like terms on the left-hand side of the equation and then adding or subtracting the same value from both sides.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the equation $x + 5x = 6x$, you can combine the like terms by adding the coefficients: $1 + 5 = 6$.

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:

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 check my solution?

A: To check your solution, you need to plug the value of the variable back into the original equation and see if it is true. If the equation is true, then 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 combining like terms
• Not adding or subtracting the same value from both sides of the equation
• Not checking the solution
• Not using the order of operations

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

• Using online resources such as Khan Academy or Mathway
• Working with a tutor or teacher
• Practicing with worksheets or exercises
• Solving real-world problems that involve linear equations

Conclusion

In this article, we have answered some frequently asked questions about solving linear equations. We have also provided some tips and resources for practicing solving linear equations. By following these tips and practicing regularly, you can become proficient in solving linear equations and apply it to a wide range of mathematical problems.

References

• [1] Khan Academy. (n.d.). Solving Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/solving-linear-equations
• [2] Math Open Reference. (n.d.). Linear Equations. Retrieved from https://www.mathopenref.com/linearequations.html

Additional Resources

• [1] Khan Academy. (n.d.). Solving Linear Equations Practice. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f4f/solving-linear-equations-practice
• [2] Mathway. (n.d.). Linear Equations Solver. Retrieved from https://www.mathway.com/linear-equations-solver