NAME: Veronica DATE: _______ PERIOD: PracticeMore Balanced Moves1. Mai And Tyler Work On The Equation $\frac{2}{5}b + 1 = -11$ Together. Mai's Solution Is $b = -25$ And Tyler's Is $b = -28$. Here Is Their

Introduction

Solving linear equations is a fundamental concept in mathematics that requires a clear understanding of algebraic operations and techniques. In this article, we will explore the process of solving linear equations, using the equation $\frac{2}{5}b + 1 = -11$ as a case study. We will examine the solutions provided by Mai and Tyler, and discuss the steps involved in solving this type of equation.

Understanding the Equation

The given equation is $\frac{2}{5}b + 1 = -11$. To solve for $b$, we need to isolate the variable $b$ on one side of the equation. The equation consists of a fraction, an integer, and a constant term. Our goal is to eliminate the fraction and the constant term, and isolate the variable $b$.

Mai's Solution

Mai's solution is $b = -25$. To verify this solution, we can substitute $b = -25$ into the original equation and check if it satisfies the equation.

$\frac{2}{5}(-25) + 1 = -11$

$-10 + 1 = -9$

$-9 \neq -11$

This shows that Mai's solution is incorrect.

Tyler's Solution

Tyler's solution is $b = -28$. To verify this solution, we can substitute $b = -28$ into the original equation and check if it satisfies the equation.

$\frac{2}{5}(-28) + 1 = -11$

$-11.2 + 1 = -10.2$

$-10.2 \neq -11$

This shows that Tyler's solution is also incorrect.

Step-by-Step Solution

To solve the equation $\frac{2}{5}b + 1 = -11$, we need to follow a series of steps:

Step 1: Subtract 1 from both sides

$\frac{2}{5}b + 1 - 1 = -11 - 1$

$\frac{2}{5}b = -12$

Step 2: Multiply both sides by 5

$\frac{2}{5}b \times 5 = -12 \times 5$

$2b = -60$

Step 3: Divide both sides by 2

$\frac{2b}{2} = \frac{-60}{2}$

$b = -30$

Step 4: Verify the solution

$\frac{2}{5}(-30) + 1 = -11$

$-12 + 1 = -11$

$-11 = -11$

This shows that the solution $b = -30$ satisfies the equation.

Conclusion

Solving linear equations requires a clear understanding of algebraic operations and techniques. In this article, we have explored the process of solving linear equations, using the equation $\frac{2}{5}b + 1 = -11$ as a case study. We have examined the solutions provided by Mai and Tyler, and discussed the steps involved in solving this type of equation. By following the step-by-step solution outlined in this article, we can solve linear equations and verify the solutions.

Common Mistakes

When solving linear equations, it is common to make mistakes such as:

• Not isolating the variable on one side of the equation
• Not eliminating the fraction and the constant term
• Not verifying the solution

To avoid these mistakes, it is essential to follow the step-by-step solution outlined in this article and to verify the solution.

Tips and Tricks

When solving linear equations, it is essential to:

• Follow the order of operations (PEMDAS)
• Use algebraic properties such as the distributive property and the commutative property
• Verify the solution

By following these tips and tricks, we can solve linear equations efficiently and accurately.

Real-World Applications

Linear equations have numerous real-world applications, such as:

• Modeling population growth and decline
• Calculating interest rates and investments
• Solving problems in physics and engineering

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

Conclusion

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Introduction

Solving linear equations is a fundamental concept in mathematics that requires a clear understanding of algebraic operations and techniques. In this article, we will explore the process of solving linear equations, using the equation $\frac{2}{5}b + 1 = -11$ as a case study. We will also provide a Q&A guide to help you understand the concepts and techniques involved in solving 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. It can be written in the form $ax + b = c$, 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. You can do this by using algebraic operations such as addition, subtraction, multiplication, and division.

Q: What are the steps involved in solving a linear equation?

A: The steps involved in solving a linear equation are:

1. Simplify the equation by combining like terms.
2. Isolate the variable on one side of the equation.
3. Use algebraic properties such as the distributive property and the commutative property.
4. Verify the solution.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, you need to combine like terms. Like terms are terms that have the same variable and coefficient. You can combine like terms by adding or subtracting their coefficients.

Q: How do I isolate the variable on one side of the equation?

A: To isolate the variable on one side of the equation, you need to use algebraic operations such as addition, subtraction, multiplication, and division. You can also use algebraic properties such as the distributive property and the commutative property.

Q: What is the distributive property?

A: The distributive property is a property of algebra that states that the product of a sum is equal to the sum of the products. It can be written as $a(b + c) = ab + ac$.

Q: What is the commutative property?

A: The commutative property is a property of algebra that states that the order of the factors does not change the result. It can be written as $ab = ba$.

Q: How do I verify the solution?

A: To verify the solution, you need to substitute the value of the variable into the original equation and check if it satisfies the equation.

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

A: Some common mistakes to avoid when solving linear equations are:

• Not isolating the variable on one side of the equation
• Not eliminating the fraction and the constant term
• Not verifying the solution

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

A: Linear equations have numerous real-world applications, such as:

• Modeling population growth and decline
• Calculating interest rates and investments
• Solving problems in physics and engineering

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

Conclusion

Solving linear equations is a fundamental concept in mathematics that requires a clear understanding of algebraic operations and techniques. In this article, we have explored the process of solving linear equations, using the equation $\frac{2}{5}b + 1 = -11$ as a case study. We have also provided a Q&A guide to help you understand the concepts and techniques involved in solving linear equations. By following the steps outlined in this article, you can solve linear equations and apply this knowledge to real-world problems.