Simplify And Solve This Equation 4m 9 5m 12 42

Simplify and Solve the Equation: 4m + 9 = 5m + 12 - 42

Solving algebraic equations is a fundamental skill in mathematics, crucial for various applications across different fields. This article will guide you through the process of simplifying and solving the equation 4m + 9 = 5m + 12 - 42, explaining each step in detail. We'll also explore related concepts to enhance your understanding of algebraic manipulation.

Understanding the Equation

Before diving into the solution, let's analyze the given equation: 4m + 9 = 5m + 12 - 42. This is a linear equation in one variable, 'm'. Our goal is to find the value of 'm' that makes the equation true. The equation involves several terms:

• 4m: This is a term with a variable (m) and a coefficient (4).
• 9: This is a constant term.
• 5m: This is another term with the variable 'm' and a coefficient (5).
• 12: This is a constant term.
• -42: This is a constant term.

Simplifying the Equation

The first step in solving this equation is to simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. In this equation, the like terms are:

• 4m and 5m: Both terms contain the variable 'm' raised to the power of 1.
• 9, 12, and -42: These are all constant terms.

Let's combine these like terms:

1. Combine the constant terms on the right-hand side: 12 - 42 = -30. The equation now becomes: 4m + 9 = 5m - 30

Solving for 'm'

Now that the equation is simplified, we can solve for 'm'. The goal is to isolate 'm' on one side of the equation. We can achieve this by performing the same operations on both sides of the equation, maintaining the equality.

Here's a step-by-step approach:

1. Subtract 4m from both sides: This eliminates 'm' from the left-hand side. The equation becomes: 9 = m - 30

2. Add 30 to both sides: This isolates 'm' on the right-hand side. The equation becomes: 9 + 30 = m

3. Simplify: 9 + 30 = 39. Therefore, m = 39

Verifying the Solution

To ensure our solution is correct, we can substitute the value of 'm' (39) back into the original equation:

4m + 9 = 5m + 12 - 42

4(39) + 9 = 5(39) + 12 - 42

156 + 9 = 195 + 12 - 42

165 = 195 - 30

165 = 165

The equation holds true, confirming that our solution, m = 39, is correct.

Expanding on Algebraic Concepts

This problem provides an opportunity to delve deeper into several key algebraic concepts:

1. The Order of Operations (PEMDAS/BODMAS)

PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) dictates the order in which operations should be performed in an equation. In our example, we simplified the right-hand side by performing subtraction before dealing with the variable terms. Understanding this order is critical for accurate equation solving.

2. Inverse Operations

Solving equations relies heavily on inverse operations. Addition and subtraction are inverse operations; multiplication and division are inverse operations. We used these inverse operations to isolate 'm': we subtracted 4m to counteract the addition of 4m, and we added 30 to counteract the subtraction of 30.

3. Properties of Equality

The solution process relies on the properties of equality:

• Addition Property of Equality: Adding the same number to both sides of an equation doesn't change its equality.
• Subtraction Property of Equality: Subtracting the same number from both sides of an equation doesn't change its equality.
• Multiplication Property of Equality: Multiplying both sides of an equation by the same non-zero number doesn't change its equality.
• Division Property of Equality: Dividing both sides of an equation by the same non-zero number doesn't change its equality.

4. Linear Equations

The given equation is a linear equation because the highest power of the variable 'm' is 1. Linear equations always represent a straight line when graphed. Understanding linear equations is essential for various mathematical applications, including modeling real-world phenomena.

Applications of Linear Equations

Linear equations have widespread applications in various fields:

• Physics: Describing motion, calculating forces, and analyzing electrical circuits.
• Engineering: Designing structures, analyzing stress and strain, and modeling fluid flow.
• Economics: Modeling supply and demand, predicting economic growth, and analyzing financial markets.
• Computer Science: Developing algorithms, creating simulations, and designing computer graphics.
• Everyday Life: Calculating distances, converting units, and budgeting expenses.

Troubleshooting Common Mistakes

When solving linear equations, several common mistakes can occur:

• Incorrect Order of Operations: Failing to follow PEMDAS/BODMAS can lead to incorrect results.
• Errors in Sign Manipulation: Mistakes in adding or subtracting negative numbers are frequent. Double-check your signs carefully!
• Incorrect Application of Inverse Operations: Forgetting to apply the same operation to both sides of the equation will lead to an incorrect solution.

Practice Makes Perfect

Solving algebraic equations efficiently and accurately requires consistent practice. Start with simpler equations and gradually increase the complexity. Work through numerous examples, focusing on understanding the underlying concepts rather than just memorizing steps. Regular practice will build your confidence and improve your problem-solving skills.

Further Exploration

To enhance your understanding further, you can explore more complex types of equations, such as quadratic equations, systems of equations, and inequalities. These more advanced topics build upon the fundamental concepts discussed in this article, providing a more comprehensive understanding of algebra. Consider utilizing online resources and textbooks to delve deeper into these subjects.

This detailed explanation of simplifying and solving the equation 4m + 9 = 5m + 12 - 42, along with the exploration of related concepts and potential pitfalls, aims to provide a thorough understanding of this fundamental mathematical skill. Remember, consistent practice and a grasp of the underlying principles are key to mastering algebra.