Solve The Equation 2(x+7)-(1-x)=12+3x
This article provides a detailed walkthrough on how to solve the algebraic equation 2(x+7)-(1-x)=12+3x. We will break down each step, explaining the underlying principles and techniques involved. Whether you're a student learning algebra or simply looking to brush up on your skills, this guide will help you understand the process and arrive at the correct solution.
Understanding the Basics of Algebraic Equations
Before diving into the specifics of solving this equation, it's crucial to grasp the fundamental concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions involve variables (usually represented by letters like 'x', 'y', or 'z'), constants (fixed numerical values), and mathematical operations (such as addition, subtraction, multiplication, and division).
The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. This value(s) is called the solution(s) or root(s) of the equation. To find the solution, we manipulate the equation using algebraic rules, aiming to isolate the variable on one side of the equation. This often involves performing the same operations on both sides of the equation to maintain the equality. Understanding these basic principles is essential for successfully tackling more complex equations, such as the one we're about to explore. Remember, algebra is a building block for higher mathematics, and a solid foundation here will benefit you greatly in the future. The process of isolating the variable is like a puzzle; each step brings you closer to revealing the value of 'x'. So, let's begin our journey into solving this equation, step by step.
Step 1: Expanding the Parentheses
Our first step in solving the equation 2(x+7)-(1-x)=12+3x is to eliminate the parentheses. This is done using the distributive property, which states that a(b+c) = ab + ac. Applying this property to the left side of the equation, we get:
2 * x + 2 * 7 - (1 - x) = 12 + 3x
This simplifies to:
2x + 14 - (1 - x) = 12 + 3x
Now, we need to address the negative sign in front of the parentheses (1 - x). Remember that a negative sign in front of parentheses means we are multiplying the entire expression inside the parentheses by -1. So, we have:
2x + 14 - 1 + x = 12 + 3x
This step is crucial because a mistake here can lead to an incorrect solution. Pay close attention to the signs when distributing the negative. Think of it as multiplying each term inside the parentheses by -1: -1 * 1 = -1 and -1 * -x = +x. By carefully expanding the parentheses, we've taken the first significant step towards simplifying the equation and making it easier to solve. The key takeaway here is the correct application of the distributive property and paying attention to the signs, especially when dealing with negative values. With this step completed, we can move on to the next stage of simplifying the equation.
Step 2: Combining Like Terms
After expanding the parentheses in the equation 2x + 14 - 1 + x = 12 + 3x, our next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have '2x' and 'x' on the left side, which are like terms, and '14' and '-1' which are also like terms.
Combining '2x' and 'x', we get:
2x + x = 3x
Combining '14' and '-1', we get:
14 - 1 = 13
So, the left side of the equation simplifies to:
3x + 13
The right side of the equation, '12 + 3x', already has its like terms combined. Therefore, our equation now looks like this:
3x + 13 = 12 + 3x
Combining like terms is a fundamental technique in algebra. It helps to simplify the equation and make it easier to manipulate. This step reduces the number of terms in the equation, bringing us closer to isolating the variable 'x'. By carefully identifying and combining like terms, we've streamlined the equation, setting the stage for the next step in our solution process. Remember, accuracy in combining like terms is paramount for arriving at the correct solution. So, always double-check your work to ensure you've correctly grouped and simplified the terms. Now, let's move on to the next stage of isolating the variable.
Step 3: Isolating the Variable
Now that we have the simplified equation 3x + 13 = 12 + 3x, our goal is to isolate the variable 'x'. This means we want to get all the terms with 'x' on one side of the equation and all the constant terms on the other side.
Let's start by subtracting '3x' from both sides of the equation. This will eliminate the '3x' term from the left side:
3x + 13 - 3x = 12 + 3x - 3x
This simplifies to:
13 = 12
Notice that the 'x' terms have canceled out completely. This is a significant observation. When the variables cancel out, we are left with a statement that is either true or false. In this case, we have:
13 = 12
This statement is false.
Isolating the variable is a critical step in solving equations. It allows us to determine the value of the variable that satisfies the equation. However, in this particular case, the variable has disappeared, leaving us with a contradiction. This indicates that the original equation has no solution. There is no value of 'x' that can make the equation true. This outcome is as important to understand as finding a solution. It demonstrates that not all equations have solutions, and recognizing such cases is a key part of problem-solving in algebra. The fact that we arrived at a false statement after correctly applying algebraic manipulations tells us definitively that the equation is inconsistent and has no solution. So, while we didn't find a numerical value for 'x', we have successfully determined the nature of the equation's solution.
Step 4: Interpreting the Result – No Solution
As we've seen in the previous step, when we tried to isolate the variable 'x' in the equation 3x + 13 = 12 + 3x, we arrived at the false statement 13 = 12. This result has a specific meaning in the context of solving equations.
When solving an equation, if the variables cancel out and you are left with a false statement, it means that the equation has no solution. This indicates that there is no value for 'x' that will make the original equation true. The two sides of the equation will never be equal, regardless of what value you substitute for 'x'.
In contrast, if the variables cancel out and you are left with a true statement (for example, 5 = 5), then the equation has infinitely many solutions. This means that any value of 'x' will satisfy the equation.
The interpretation of the result is a vital part of the problem-solving process. It's not enough to just perform the algebraic manipulations; you also need to understand what the outcome means. In this case, the fact that we have no solution tells us something important about the relationship between the expressions on either side of the original equation. They are fundamentally incompatible, and there's no 'x' that can bridge that gap. Understanding these nuances is what elevates your algebraic skills from mere computation to true problem-solving ability. Recognizing when an equation has no solution is just as valuable as finding a solution, as it prevents you from wasting time searching for an answer that doesn't exist. This understanding highlights the importance of careful analysis and interpretation in mathematics.
Conclusion
In this comprehensive guide, we've walked through the process of solving the equation 2(x+7)-(1-x)=12+3x. We began by understanding the basics of algebraic equations and the goal of isolating the variable. We then systematically worked through the steps:
- Expanding the parentheses using the distributive property.
- Combining like terms on each side of the equation.
- Attempting to isolate the variable 'x'.
Through these steps, we discovered that the equation simplifies to a false statement: 13 = 12. This led us to the crucial conclusion that the equation has no solution. There is no value of 'x' that can satisfy the equation.
This exercise highlights the importance of not just knowing the algebraic techniques but also understanding how to interpret the results. It's a reminder that not all equations have solutions, and recognizing such cases is a key skill in algebra and mathematics in general. The ability to methodically solve equations and accurately interpret the outcomes is essential for success in mathematics and related fields. This example provides a valuable lesson in problem-solving, demonstrating that a clear, step-by-step approach, combined with careful interpretation, can lead to a complete understanding of the problem, even when the solution is "no solution." By mastering these skills, you'll be well-equipped to tackle a wide range of algebraic challenges and deepen your mathematical understanding.