Solving For X In The Equation 2(x-3) + 9 = 3(x+1) + X
In the realm of algebra, solving equations is a fundamental skill. This article delves into the process of finding the value of the unknown variable, x, in the equation 2(x-3) + 9 = 3(x+1) + x. This equation, while seemingly complex at first glance, can be systematically solved using the principles of algebraic manipulation. By understanding the steps involved in isolating x, we can gain a deeper appreciation for the power and elegance of mathematical problem-solving.
Unraveling the Equation: A Step-by-Step Approach
The task of finding the value of x requires a methodical approach. We begin by simplifying both sides of the equation, removing parentheses, and combining like terms. This sets the stage for isolating x on one side of the equation, allowing us to determine its numerical value. This process not only solves the specific equation but also reinforces essential algebraic techniques applicable to a wide range of mathematical problems.
1. Expanding the Parentheses: The First Key Step
The initial step in solving the equation 2(x-3) + 9 = 3(x+1) + x involves expanding the parentheses on both sides. This is achieved by applying the distributive property, which states that a(b + c) = ab + ac. On the left side, we multiply 2 by both x and -3, resulting in 2x - 6. On the right side, we multiply 3 by both x and 1, resulting in 3x + 3. The equation now transforms into:
2x - 6 + 9 = 3x + 3 + x
This expansion eliminates the parentheses, making the equation easier to work with. It's a crucial step because it allows us to combine like terms in the next stage of the solution.
2. Combining Like Terms: Simplifying the Equation
After expanding the parentheses, the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (e.g., 2x and 3x) or are constants (e.g., -6 and 9). On the left side, we have the constants -6 and 9, which combine to give +3. On the right side, we have the terms 3x and x, which combine to give 4x. The equation now simplifies to:
2x + 3 = 4x + 3
This simplification makes the equation more manageable and brings us closer to isolating the variable x.
3. Isolating the Variable: Moving Towards the Solution
To isolate x, we need to get all the x terms on one side of the equation and all the constant terms on the other. A common strategy is to subtract the smaller x term from both sides. In this case, we subtract 2x from both sides of the equation:
2x + 3 - 2x = 4x + 3 - 2x
This simplifies to:
3 = 2x + 3
Now, we need to isolate the term with x by subtracting 3 from both sides:
3 - 3 = 2x + 3 - 3
This further simplifies to:
0 = 2x
4. Solving for x: The Final Step
The final step in solving for x is to divide both sides of the equation by the coefficient of x, which is 2 in this case:
0 / 2 = 2x / 2
This gives us:
0 = x
Therefore, the solution to the equation 2(x-3) + 9 = 3(x+1) + x is x = 0. This means that if we substitute 0 for x in the original equation, both sides of the equation will be equal.
Verifying the Solution: Ensuring Accuracy
After finding a solution, it's always a good practice to verify its accuracy. This is done by substituting the value of x back into the original equation and checking if both sides are equal. Substituting x = 0 into the equation 2(x-3) + 9 = 3(x+1) + x, we get:
2(0-3) + 9 = 3(0+1) + 0
Simplifying the left side:
2(-3) + 9 = -6 + 9 = 3
Simplifying the right side:
3(1) + 0 = 3 + 0 = 3
Since both sides of the equation equal 3, our solution of x = 0 is verified to be correct. This verification step confirms that we have accurately solved the equation.
The Significance of Algebraic Manipulation
The process of solving the equation 2(x-3) + 9 = 3(x+1) + x highlights the importance of algebraic manipulation. By applying the distributive property, combining like terms, and isolating the variable, we systematically transformed a complex equation into a simple solution. These techniques are fundamental to algebra and are used extensively in various mathematical and scientific disciplines. Mastering these skills provides a strong foundation for tackling more advanced mathematical problems.
Mastering the Distributive Property: A Cornerstone of Algebra
The distributive property, as demonstrated in the expansion of parentheses, is a cornerstone of algebra. It allows us to simplify expressions and equations by multiplying a term across a sum or difference. This property is essential for manipulating algebraic expressions and is frequently used in solving equations, simplifying fractions, and working with polynomials. A strong understanding of the distributive property is crucial for success in algebra and beyond.
Combining Like Terms: Streamlining Algebraic Expressions
Combining like terms is another essential algebraic skill. It involves identifying terms with the same variable raised to the same power and adding or subtracting their coefficients. This process streamlines algebraic expressions, making them easier to work with and understand. Combining like terms is a fundamental step in solving equations, simplifying expressions, and performing other algebraic operations.
Isolating the Variable: The Key to Solving Equations
Isolating the variable is the ultimate goal in solving equations. It involves manipulating the equation to get the variable by itself on one side. This is achieved by performing the same operations on both sides of the equation, ensuring that the equation remains balanced. Isolating the variable allows us to determine its value and find the solution to the equation. This skill is fundamental to algebra and is used in a wide range of mathematical applications.
Conclusion: The Power of Algebraic Problem-Solving
In conclusion, the value of x in the equation 2(x-3) + 9 = 3(x+1) + x is 0. This solution was found by systematically applying algebraic principles, including expanding parentheses, combining like terms, and isolating the variable. The process of solving this equation demonstrates the power and elegance of algebraic problem-solving. By mastering these techniques, individuals can confidently tackle a wide range of mathematical challenges. The ability to manipulate equations and solve for unknowns is a valuable skill in various fields, from science and engineering to finance and economics. The principles learned in solving this equation serve as a foundation for more advanced mathematical concepts and applications.
This exploration into solving for x underscores the beauty and practicality of algebra. It's not just about finding a number; it's about understanding the underlying principles and developing a logical approach to problem-solving. These skills are transferable and will serve you well in many aspects of life, fostering critical thinking and analytical abilities.
