Explanation of the Correct Solution

Oct 13, 2025 by magicalceylontours.com 68 views

Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving these equations to arrive at the correct solution. In this article, we will examine a given calculation for solving for x and identify the mistake in the process. We will then provide the correct solution to the equation.

The Given Calculation

The given calculation is as follows:

{ \begin{aligned} 6x + 2x + 8 & = 180 \\ \frac{16x}{16} & = \frac{180}{16} \\ x & = 11.25 \end{aligned} \}

Identifying the Mistake

The mistake in the given calculation lies in the second step, where the equation is simplified by dividing both sides by 16. However, this simplification is incorrect because it ignores the constant term 8 in the original equation.

Correcting the Mistake

To correct the mistake, we need to follow the correct order of operations and simplify the equation step by step. The correct steps are as follows:

{ \begin{aligned} 6x + 2x + 8 & = 180 \\ 8x + 8 & = 180 \\ 8x & = 172 \\ x & = \frac{172}{8} \\ x & = 21.5 \end{aligned} \}

Explanation of the Correct Solution

In the correct solution, we first combine the like terms 6x and 2x to get 8x. We then subtract 8 from both sides of the equation to isolate the term 8x. This gives us 8x = 172. Finally, we divide both sides of the equation by 8 to solve for x, which gives us x = 21.5.

Conclusion

In conclusion, the mistake in the given calculation was ignoring the constant term 8 in the original equation. By following the correct order of operations and simplifying the equation step by step, we arrived at the correct solution, which is x = 21.5.

Importance of Following the Order of Operations

Following the order of operations is crucial in solving linear equations. It ensures that we simplify the equation correctly and avoid mistakes. In this case, ignoring the constant term 8 led to an incorrect solution. By following the correct order of operations, we can ensure that we arrive at the correct solution.

Real-World Applications of Solving Linear Equations

Solving linear equations has numerous real-world applications. In finance, linear equations are used to calculate interest rates and investment returns. In science, linear equations are used to model population growth and chemical reactions. In engineering, linear equations are used to design and optimize systems.

Tips for Solving Linear Equations

Here are some tips for solving linear equations:

Follow the order of operations: Simplify the equation step by step, following the order of operations (PEMDAS).
Combine like terms: Combine like terms to simplify the equation.
Isolate the variable: Isolate the variable by subtracting or adding the same value to both sides of the equation.
Check your solution: Check your solution by plugging it back into the original equation.

By following these tips and practicing solving linear equations, you can become proficient in solving these equations and apply them to real-world problems.

Conclusion

In conclusion, solving linear equations is a fundamental concept in mathematics that has numerous real-world applications. By following the correct order of operations and simplifying the equation step by step, we can arrive at the correct solution. By practicing solving linear equations and following the tips provided, you can become proficient in solving these equations and apply them to real-world problems.

Introduction

In our previous article, we examined a given calculation for solving for x and identified the mistake in the process. We then provided the correct solution to the equation. In this article, we will answer some frequently asked questions related to solving linear equations.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The acronym PEMDAS is commonly used to remember the order of operations:

P: Parentheses - Evaluate expressions inside parentheses first.
E: Exponents - Evaluate any exponential expressions next.
M: Multiplication and Division - Evaluate any multiplication and division operations from left to right.
A: 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, we need to identify the terms that have the same variable and coefficient. We can then add or subtract these terms to simplify the equation. For example, in the equation 2x + 3x + 4, we can combine the like terms 2x and 3x to get 5x.

Q: How do I isolate the variable?

A: To isolate the variable, we need to get the variable term by itself on one side of the equation. We can do this by adding or subtracting the same value to both sides of the equation. For example, in the equation 2x + 3 = 5, we can subtract 3 from both sides to get 2x = 2.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, the equation 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, the equation x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I check my solution?

A: To check your solution, you need to plug it back into the original equation and see if it is true. For example, if you solve the equation 2x + 3 = 5 and get x = 1, you can plug x = 1 back into the original equation to see if it is true: 2(1) + 3 = 5. If this 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:

Ignoring the constant term: Make sure to include the constant term in your equation and simplify it correctly.
Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying the equation.
Not isolating the variable: Make sure to isolate the variable by adding or subtracting the same value to both sides of the equation.
Not checking your solution: Make sure to check your solution by plugging it back into the original equation.