Practice: More Balanced MovesMai 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 Work:Mai's Work:$\[ \begin{align*} \frac{2}{5} B + 1

Introduction

Solving linear equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we will explore the process of solving linear equations, using the example of Mai and Tyler working together to solve the equation $\frac{2}{5} b + 1 = -11$.

Understanding Linear Equations

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. Linear equations can be solved using various methods, including addition, subtraction, multiplication, and division.

Mai's Solution

Mai's solution to the equation $\frac{2}{5} b + 1 = -11$ is $b = -25$. To arrive at this solution, Mai likely followed these steps:

1. Isolate the variable: Mai started by isolating the variable $b$ on one side of the equation. This can be done by subtracting 1 from both sides of the equation, resulting in $\frac{2}{5} b = -12$.
2. Multiply both sides by the reciprocal: To get rid of the fraction, Mai multiplied both sides of the equation by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$. This resulted in $b = -12 \times \frac{5}{2}$.
3. Simplify the expression: Mai simplified the expression by multiplying $-12$ by $\frac{5}{2}$, resulting in $b = -30$.

However, Mai's solution is incorrect. Let's re-examine the steps and find the correct solution.

Tyler's Solution

Tyler's solution to the equation $\frac{2}{5} b + 1 = -11$ is $b = -28$. To arrive at this solution, Tyler likely followed these steps:

1. Isolate the variable: Tyler started by isolating the variable $b$ on one side of the equation. This can be done by subtracting 1 from both sides of the equation, resulting in $\frac{2}{5} b = -12$.
2. Multiply both sides by the reciprocal: To get rid of the fraction, Tyler multiplied both sides of the equation by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$. This resulted in $b = -12 \times \frac{5}{2}$.
3. Simplify the expression: Tyler simplified the expression by multiplying $-12$ by $\frac{5}{2}$, resulting in $b = -30$.

However, Tyler's solution is also incorrect. Let's re-examine the steps and find the correct solution.

Correct Solution

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

1. Isolate the variable: Subtract 1 from both sides of the equation, resulting in $\frac{2}{5} b = -12$.
2. Multiply both sides by the reciprocal: Multiply both sides of the equation by the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$. This resulted in $b = -12 \times \frac{5}{2}$.
3. Simplify the expression: Simplify the expression by multiplying $-12$ by $\frac{5}{2}$, resulting in $b = -30$.

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Introduction

Solving linear equations is a fundamental concept in mathematics that involves finding the value of a variable that makes an equation true. In this article, we will explore the process of solving linear equations, using the example of Mai and Tyler working together to solve the equation $\frac{2}{5} b + 1 = -11$.

Q&A

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. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to isolate the variable on one side of the equation. This can be done by subtracting or adding the same value to both sides of the equation.

Q: How do I get rid of a fraction in a linear equation?

A: To get rid of a fraction in a linear equation, you need to multiply both sides of the equation by the reciprocal of the fraction.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.

Q: How do I simplify an expression in a linear equation?

A: To simplify an expression in a linear equation, you need to combine like terms and perform any necessary arithmetic operations.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x$ and $3x$ are like terms.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug the value of the variable back into the original equation and verify that it is true.

Q: What if I get a different solution to a linear equation?

A: If you get a different solution to a linear equation, it may be because you made a mistake in your calculations or because the equation has multiple solutions.

Q: How do I know if an equation has multiple solutions?

A: An equation has multiple solutions if it is a quadratic equation or if it has multiple variables.

Q: What is a quadratic equation?

A: A quadratic equation is an equation in which the highest power of the variable is 2. It 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 solve a quadratic equation?

A: To solve a quadratic equation, you need to use the quadratic formula or factor the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What if I get stuck on a linear equation?

A: If you get stuck on a linear equation, you can try using a different method or seeking help from a teacher or tutor.

Q: How do I know if I have solved a linear equation correctly?

A: You can check your solution to a linear equation by plugging the value of the variable back into the original equation and verifying that it is true.

Q: What if I make a mistake on a linear equation?

A: If you make a mistake on a linear equation, you can try to correct it by re-reading the equation and re-working the solution.

Q: How do I avoid making mistakes on linear equations?

A: To avoid making mistakes on linear equations, you need to carefully read the equation and follow the correct steps to solve it.

Q: What if I am unsure about a linear equation?

A: If you are unsure about a linear equation, you can try to ask a teacher or tutor for help or seek additional resources to learn more about linear equations.

Q: How do I know if I have learned enough about linear equations?

A: You can know if you have learned enough about linear equations by practicing solving linear equations and checking your solutions to ensure that they are correct.

Q: What if I need to solve a linear equation with multiple variables?

A: If you need to solve a linear equation with multiple variables, you can try to use a different method or seek help from a teacher or tutor.

Q: How do I know if I have solved a linear equation with multiple variables correctly?

A: You can check your solution to a linear equation with multiple variables by plugging the values of the variables back into the original equation and verifying that it is true.

Q: What if I make a mistake on a linear equation with multiple variables?

A: If you make a mistake on a linear equation with multiple variables, you can try to correct it by re-reading the equation and re-working the solution.

Q: How do I avoid making mistakes on linear equations with multiple variables?

A: To avoid making mistakes on linear equations with multiple variables, you need to carefully read the equation and follow the correct steps to solve it.

Q: What if I am unsure about a linear equation with multiple variables?

A: If you are unsure about a linear equation with multiple variables, you can try to ask a teacher or tutor for help or seek additional resources to learn more about linear equations.

Q: How do I know if I have learned enough about linear equations with multiple variables?

A: You can know if you have learned enough about linear equations with multiple variables by practicing solving linear equations with multiple variables and checking your solutions to ensure that they are correct.

Q: What if I need to solve a linear equation with a fraction?

A: If you need to solve a linear equation with a fraction, you can try to use a different method or seek help from a teacher or tutor.

Q: How do I know if I have solved a linear equation with a fraction correctly?

A: You can check your solution to a linear equation with a fraction by plugging the value of the variable back into the original equation and verifying that it is true.

Q: What if I make a mistake on a linear equation with a fraction?

A: If you make a mistake on a linear equation with a fraction, you can try to correct it by re-reading the equation and re-working the solution.

Q: How do I avoid making mistakes on linear equations with fractions?

A: To avoid making mistakes on linear equations with fractions, you need to carefully read the equation and follow the correct steps to solve it.

Q: What if I am unsure about a linear equation with a fraction?

A: If you are unsure about a linear equation with a fraction, you can try to ask a teacher or tutor for help or seek additional resources to learn more about linear equations.

Q: How do I know if I have learned enough about linear equations with fractions?

A: You can know if you have learned enough about linear equations with fractions by practicing solving linear equations with fractions and checking your solutions to ensure that they are correct.

Q: What if I need to solve a linear equation with a decimal?

A: If you need to solve a linear equation with a decimal, you can try to use a different method or seek help from a teacher or tutor.

Q: How do I know if I have solved a linear equation with a decimal correctly?

A: You can check your solution to a linear equation with a decimal by plugging the value of the variable back into the original equation and verifying that it is true.

Q: What if I make a mistake on a linear equation with a decimal?

A: If you make a mistake on a linear equation with a decimal, you can try to correct it by re-reading the equation and re-working the solution.

Q: How do I avoid making mistakes on linear equations with decimals?

A: To avoid making mistakes on linear equations with decimals, you need to carefully read the equation and follow the correct steps to solve it.

Q: What if I am unsure about a linear equation with a decimal?

A: If you are unsure about a linear equation with a decimal, you can try to ask a teacher or tutor for help or seek additional resources to learn more about linear equations.

Q: How do I know if I have learned enough about linear equations with decimals?

A: You can know if you have learned enough about linear equations with decimals by practicing solving linear equations with decimals and checking your solutions to ensure that they are correct.

Q: What if I need to solve a linear equation with a negative number?

A: If you need to solve a linear equation with a negative number, you can try to use a different method or seek help from a teacher or tutor.

Q: How do I know if I have solved a linear equation with a negative number correctly?

A: You can check your solution to a linear equation with a negative number by plugging the value of the variable back into the original equation and verifying that it is true.

Q: What if I make a mistake on a linear equation with a negative number?

A: If you make a mistake on