Describe The Steps You Used To Solve The Equation And Find The Amount Of Carrie's Allowance.Linear Equation: $\frac{1}{4} A+\frac{1}{3} A+8=22$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is an essential skill for students to master. In this article, we will walk through the steps to solve a linear equation and find the amount of Carrie's allowance. The equation we will be working with is $\frac{1}{4} a+\frac{1}{3} a+8=22$. We will break down the solution into manageable steps, making it easy to follow and understand.

Step 1: Write Down the Equation

The given equation is $\frac{1}{4} a+\frac{1}{3} a+8=22$. The first step is to write down the equation and identify the variables and constants involved.

Variables and Constants

• Variables: $a$
• Constants: $\frac{1}{4}$, $\frac{1}{3}$, 8, 22

Step 2: Combine Like Terms

The next step is to combine like terms, which involves adding or subtracting terms that have the same variable. In this case, we can combine the two terms with the variable $a$.

Combining Like Terms

$\frac{1}{4} a+\frac{1}{3} a = \frac{3}{12} a + \frac{4}{12} a = \frac{7}{12} a$

The equation now becomes:

$\frac{7}{12} a + 8 = 22$

Step 3: Isolate the Variable

The goal is to isolate the variable $a$ on one side of the equation. To do this, we need to get rid of the constant term 8. We can do this by subtracting 8 from both sides of the equation.

Isolating the Variable

$\frac{7}{12} a + 8 - 8 = 22 - 8$

$\frac{7}{12} a = 14$

Step 4: Solve for the Variable

Now that we have isolated the variable, we can solve for $a$ by multiplying both sides of the equation by the reciprocal of $\frac{7}{12}$, which is $\frac{12}{7}$.

Solving for the Variable

$\frac{12}{7} \cdot \frac{7}{12} a = \frac{12}{7} \cdot 14$

$a = \frac{12}{7} \cdot 14$

$a = 24$

Conclusion

In this article, we walked through the steps to solve a linear equation and find the amount of Carrie's allowance. The equation we worked with was $\frac{1}{4} a+\frac{1}{3} a+8=22$. We combined like terms, isolated the variable, and solved for $a$. The final answer is $a = 24$. This problem demonstrates the importance of following the order of operations and using algebraic techniques to solve linear equations.

Real-World Applications

Linear equations have numerous real-world applications, including finance, science, and engineering. For example, in finance, linear equations can be used to calculate interest rates, investment returns, and loan payments. In science, linear equations can be used to model population growth, chemical reactions, and physical systems. In engineering, linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations and use algebraic techniques to isolate the variable. Here are some tips and tricks to help you solve linear equations:

• Use a systematic approach: Break down the problem into smaller steps and follow a systematic approach to solve the equation.
• Combine like terms: Combine terms with the same variable to simplify the equation.
• Isolate the variable: Get rid of the constant term by subtracting or adding it to both sides of the equation.
• Use algebraic techniques: Use techniques such as multiplication, division, addition, and subtraction to solve for the variable.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) 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 techniques such as addition, subtraction, multiplication, and division.

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 combine like terms?

A: To combine like terms, you need to add or subtract terms that have the same variable. For example, if you have the equation 2x + 3x, you can combine the like terms by adding them together: 2x + 3x = 5x.

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get rid of the constant term by subtracting or adding it to both sides of the equation. For example, if you have the equation x + 2 = 5, you can isolate the variable by subtracting 2 from both sides: x + 2 - 2 = 5 - 2, which simplifies to x = 3.

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
• Making errors when adding, subtracting, multiplying, or dividing

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. For example, if you solved the equation x + 2 = 5 and got x = 3, you can plug x = 3 back into the original equation: 3 + 2 = 5, which is true.

Q: What are some real-world applications of linear equations?

A: Linear equations have numerous real-world applications, including:

• Finance: Linear equations can be used to calculate interest rates, investment returns, and loan payments.
• Science: Linear equations can be used to model population growth, chemical reactions, and physical systems.
• Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve linear equations.