Solve For { M $} . . . { 3 + 4(m + 7) = 57 \} { M = $}$

$$

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 focus on solving a specific type of linear equation, where the variable $m$ is isolated on one side of the equation. We will use the given equation $3 + 4(m + 7) = 57$ as an example to demonstrate the step-by-step process of solving for $m$.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a linear equation, which means it is an equation in which the highest power of the variable (in this case, $m$) is 1. The equation is also a multi-step equation, which means it requires multiple steps to solve.

The equation is given as $3 + 4(m + 7) = 57$. To solve for $m$, we need to isolate the variable $m$ on one side of the equation. We can start by simplifying the left-hand side of the equation.

Simplifying the Left-Hand Side

To simplify the left-hand side of the equation, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. In this case, we can apply the distributive property to the term $4(m + 7)$.

# Simplifying the left-hand side of the equation
import sympy as sp
m = sp.symbols('m')

equation = 3 + 4*(m + 7) - 57

simplified_equation = sp.simplify(equation)
print(simplified_equation)

The simplified equation is $4m + 28 + 3 = 57$. We can further simplify this equation by combining like terms.

Combining Like Terms

To combine like terms, we need to identify the terms that have the same variable and coefficient. In this case, we have two terms with the variable $m$, which are $4m$ and $28$. We can combine these terms by adding their coefficients.

# Combining like terms
import sympy as sp

m = sp.symbols('m')

simplified_equation = 4*m + 31 - 57

combined_equation = sp.simplify(simplified_equation)
print(combined_equation)

The combined equation is $4m - 26 = 0$. We can now solve for $m$ by isolating the variable on one side of the equation.

Solving for $m$

To solve for $m$, we need to isolate the variable on one side of the equation. We can do this by adding $26$ to both sides of the equation.

# Solving for m
import sympy as sp

m = sp.symbols('m')

combined_equation = 4*m - 26

solved_equation = sp.Eq(combined_equation + 26, 26)
print(solved_equation)

The solved equation is $4m = 26$. We can now solve for $m$ by dividing both sides of the equation by $4$.

Final Solution

To solve for $m$, we need to divide both sides of the equation by $4$. This will give us the value of $m$.

# Final solution
import sympy as sp

m = sp.symbols('m')

solved_equation = 4*m - 26

final_solution = sp.Eq(solved_equation / 4, 26 / 4)
print(final_solution)

The final solution is $m = \frac{26}{4}$. We can simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is $2$.

Simplifying the Final Solution

To simplify the final solution, we need to divide both the numerator and denominator by their greatest common divisor, which is $2$.

# Simplifying the final solution
import sympy as sp

m = sp.symbols('m')

final_solution = 26 / 4

simplified_solution = sp.simplify(final_solution)
print(simplified_solution)

The simplified solution is $m = \frac{13}{2}$. This is the final answer to the equation $3 + 4(m + 7) = 57$.

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation, where the variable $m$ is isolated on one side of the equation. We have used the given equation $3 + 4(m + 7) = 57$ as an example to illustrate the process. We have simplified the left-hand side of the equation, combined like terms, and solved for $m$ by isolating the variable on one side of the equation. The final solution is $m = \frac{13}{2}$.

Frequently Asked Questions

• What is a linear equation? A linear equation is an equation in which the highest power of the variable is 1.
• How do I simplify the left-hand side of a linear equation? To simplify the left-hand side of a linear equation, you need to apply the distributive property and combine like terms.
• How do I solve for $m$ in a linear equation? To solve for $m$ in a linear equation, you need to isolate the variable on one side of the equation by adding or subtracting the same value from both sides of the equation.

References

• [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1d7d7d/x2f1d7d7d/x2f1d7d7d
• [2] Mathway. (n.d.). Linear Equations. Retrieved from https://www.mathway.com/subjects/linear-equations

Additional Resources

• [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1d7d7d/x2f1d7d7d/x2f1d7d7d
• [2] Mathway. (n.d.). Linear Equations. Retrieved from https://www.mathway.com/subjects/linear-equations

Glossary

• Linear Equation: An equation in which the highest power of the variable is 1.
• Distributive Property: A property of arithmetic that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.
• Like Terms: Terms that have the same variable and coefficient.
• Isolate the Variable: To isolate the variable on one side of the equation by adding or subtracting the same value from both sides of the equation.
  

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we demonstrated the step-by-step process of solving a linear equation, where the variable $m$ is isolated on one side of the equation. In this article, we will provide a Q&A guide to help students understand and solve linear equations.

Q&A Guide

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1.

Q: How do I simplify the left-hand side of a linear equation?

A: To simplify the left-hand side of a linear equation, you need to apply the distributive property and combine like terms.

Q: How do I solve for $m$ in a linear equation?

A: To solve for $m$ in a linear equation, you need to isolate the variable on one side of the equation by adding or subtracting the same value from both sides of the equation.

Q: What is the distributive property?

A: The distributive property is a property of arithmetic that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Q: What are like terms?

A: Like terms are terms that have the same variable and coefficient.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which mathematical operations should be performed. The order of operations is:

1. Parentheses
2. Exponents
3. Multiplication and Division
4. Addition and Subtraction

Q: How do I use the order of operations to solve a linear equation?

A: To use the order of operations to solve a linear equation, you need to follow the order of operations and perform the operations in the correct order.

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 combining like terms
• Not isolating the variable on one side of the equation
• Not checking the solution for accuracy

Q: How do I check the solution for accuracy?

A: To check the solution for accuracy, you need to plug the solution back into the original equation and verify that it is true.

Example Questions

Q: Solve for $m$ in the equation $2m + 5 = 11$.

A: To solve for $m$, we need to isolate the variable on one side of the equation. We can do this by subtracting 5 from both sides of the equation.

$2m + 5 - 5 = 11 - 5$

$2m = 6$

$m = \frac{6}{2}$

$m = 3$

Q: Solve for $m$ in the equation $m - 2 = 7$.

A: To solve for $m$, we need to isolate the variable on one side of the equation. We can do this by adding 2 to both sides of the equation.

$m - 2 + 2 = 7 + 2$

$m = 9$

Conclusion

In this article, we have provided a Q&A guide to help students understand and solve linear equations. We have covered topics such as the distributive property, like terms, and the order of operations. We have also provided example questions to help students practice solving linear equations.

Frequently Asked Questions

• What is a linear equation?
• How do I simplify the left-hand side of a linear equation?
• How do I solve for $m$ in a linear equation?
• What is the distributive property?
• What are like terms?
• How do I combine like terms?
• What is the order of operations?
• How do I use the order of operations to solve a linear equation?
• What are some common mistakes to avoid when solving linear equations?
• How do I check the solution for accuracy?

References

• [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1d7d7d/x2f1d7d7d/x2f1d7d7d
• [2] Mathway. (n.d.). Linear Equations. Retrieved from https://www.mathway.com/subjects/linear-equations

Additional Resources

• [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1d7d7d/x2f1d7d7d/x2f1d7d7d
• [2] Mathway. (n.d.). Linear Equations. Retrieved from https://www.mathway.com/subjects/linear-equations

Glossary

• Linear Equation: An equation in which the highest power of the variable is 1.
• Distributive Property: A property of arithmetic that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.
• Like Terms: Terms that have the same variable and coefficient.
• Isolate the Variable: To isolate the variable on one side of the equation by adding or subtracting the same value from both sides of the equation.
• Order of Operations: A set of rules that dictate the order in which mathematical operations should be performed.