Solving 4m + 9 + 5m - 12 = 42 A Step By Step Guide
In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. To solve an equation, we aim to find the value(s) of the variable(s) that make the equation true. This article delves into the process of simplifying and solving a linear equation, providing a step-by-step guide to arrive at the correct solution. We will specifically focus on the equation 4m + 9 + 5m - 12 = 42, demonstrating how to combine like terms, isolate the variable, and ultimately determine the value of 'm' that satisfies the equation. This process is crucial not only for academic pursuits but also for various real-world applications where mathematical modeling and problem-solving are essential.
Before we dive into the solution, let's briefly discuss linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable's exponent is implicitly 1. Linear equations are characterized by their straightforward nature and the fact that they represent a straight line when graphed on a coordinate plane. Solving linear equations typically involves a series of algebraic manipulations aimed at isolating the variable on one side of the equation. This often involves combining like terms, applying the properties of equality (such as addition, subtraction, multiplication, and division), and simplifying the expression until the variable is alone. Understanding these fundamental concepts is crucial for tackling more complex mathematical problems in the future. This article provides a clear and concise approach to solving linear equations, making it an invaluable resource for students and anyone looking to refresh their algebra skills.
1. Combine Like Terms
The first step in simplifying the equation 4m + 9 + 5m - 12 = 42 is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In this equation, we have two terms with the variable 'm' (4m and 5m) and two constant terms (9 and -12). Combining these terms, we get:
(4m + 5m) + (9 - 12) = 42
This simplifies to:
9m - 3 = 42
Combining like terms is a fundamental step in solving algebraic equations. It allows us to consolidate the equation and make it easier to manipulate. By grouping similar terms together, we reduce the complexity of the equation and pave the way for isolating the variable. This process is not only applicable to linear equations but also to more complex polynomial equations and algebraic expressions. Mastering the art of combining like terms is essential for success in algebra and beyond. This step-by-step approach ensures clarity and understanding, making it easier for readers to follow along and grasp the underlying principles of equation solving.
2. Isolate the Variable Term
Now that we have simplified the equation to 9m - 3 = 42, our next goal is to isolate the term containing the variable 'm'. To do this, we need to eliminate the constant term (-3) from the left side of the equation. We can achieve this by adding 3 to both sides of the equation. This is based on the fundamental property of equality, which states that adding the same value to both sides of an equation maintains the equality.
9m - 3 + 3 = 42 + 3
This simplifies to:
9m = 45
Isolating the variable term is a crucial step in solving equations. It brings us closer to determining the value of the variable that satisfies the equation. By applying the properties of equality, we ensure that the equation remains balanced while we manipulate it to isolate the variable. This step demonstrates the importance of understanding and applying algebraic principles to solve mathematical problems. The clarity of this explanation helps readers understand the reasoning behind each step, fostering a deeper understanding of the equation-solving process.
3. Solve for the Variable
We've now arrived at the equation 9m = 45. To solve for 'm', we need to isolate 'm' by dividing both sides of the equation by the coefficient of 'm', which is 9. Again, this is based on the property of equality, which allows us to divide both sides of an equation by the same non-zero value without changing the equality.
(9m) / 9 = 45 / 9
This simplifies to:
m = 5
Therefore, the solution to the equation 4m + 9 + 5m - 12 = 42 is m = 5. This final step demonstrates the culmination of the previous steps, where we systematically simplified and manipulated the equation to isolate the variable and find its value. Solving for the variable is the ultimate goal of equation solving, and this clear and concise explanation helps readers understand how to achieve this goal. The step-by-step approach makes it easy for readers to follow along and grasp the underlying principles of algebraic manipulation.
To ensure the accuracy of our solution, it's always a good practice to verify the answer by substituting the value of 'm' back into the original equation. Let's substitute m = 5 into the equation 4m + 9 + 5m - 12 = 42:
4(5) + 9 + 5(5) - 12 = 42
Simplifying the left side of the equation, we get:
20 + 9 + 25 - 12 = 42
42 = 42
Since the left side of the equation equals the right side, our solution m = 5 is correct. Verifying the solution is a crucial step in the problem-solving process. It provides confidence in the accuracy of the answer and helps identify any potential errors in the calculations. This step reinforces the importance of thoroughness and attention to detail in mathematics. The clear and concise explanation of the verification process makes it easy for readers to understand and implement this practice in their own problem-solving endeavors.
In this article, we have successfully simplified and solved the equation 4m + 9 + 5m - 12 = 42. By combining like terms, isolating the variable term, and solving for the variable, we determined that the solution is m = 5. We also verified our solution by substituting it back into the original equation. This step-by-step approach provides a clear and concise method for solving linear equations, a fundamental skill in mathematics. Mastering this process not only helps in academic settings but also in various real-world applications where mathematical problem-solving is essential. The ability to confidently solve equations is a valuable asset in many fields, from engineering and finance to science and technology. This article serves as a comprehensive guide for anyone looking to improve their equation-solving skills and deepen their understanding of algebraic principles.
The final answer is A. m = 5
