Solving 7(2x+3)+8(4x-5)=12(2x+5)+9 A Step-by-Step Guide
Solve the equation 7(2x+3)+8(4x-5)=12(2x+5)+9.
In the realm of algebra, equations serve as the cornerstone for problem-solving and mathematical reasoning. Among the diverse types of equations, linear equations hold a prominent position due to their simplicity and widespread applicability. This article delves into the intricacies of solving a specific linear equation: 7(2x+3)+8(4x-5)=12(2x+5)+9. We will embark on a step-by-step journey, meticulously dissecting each stage of the solution process, while also emphasizing the fundamental principles that underpin the algebraic manipulations involved.
1. The Foundation: Understanding Linear Equations
Before we delve into the specific equation at hand, it is crucial to establish a solid understanding of linear equations. A linear equation is an algebraic equation in which the highest power of the variable is 1. In simpler terms, the variable appears only once in each term, and there are no exponents or other complex functions applied to the variable. Linear equations can be represented in the general form: ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable.
Linear equations are fundamental because they model a wide array of real-world phenomena, from simple financial calculations to complex scientific relationships. Their solutions provide insights into the values that satisfy the equation, enabling us to make predictions and draw conclusions. Mastering the art of solving linear equations is, therefore, an essential skill for anyone venturing into the realms of mathematics, science, or engineering.
2. The Equation Unveiled: 7(2x+3)+8(4x-5)=12(2x+5)+9
Now, let's turn our attention to the specific equation that forms the focus of this article: 7(2x+3)+8(4x-5)=12(2x+5)+9. This equation, while seemingly complex at first glance, is essentially a linear equation disguised within parentheses and multiple terms. Our mission is to unravel this equation, simplify it, and ultimately isolate the variable 'x' to determine its value.
This equation presents a common scenario in algebra: the presence of parentheses that obscure the underlying linear structure. The key to unlocking the equation lies in the strategic application of the distributive property, a fundamental principle that allows us to multiply a term outside the parentheses by each term inside the parentheses. This process will effectively eliminate the parentheses and pave the way for further simplification.
3. Step 1: Distribute and Conquer
The first step in solving our equation involves the strategic application of the distributive property. This property, a cornerstone of algebraic manipulation, allows us to multiply a term outside a set of parentheses by each term inside the parentheses. By carefully distributing the constants, we can effectively eliminate the parentheses and reveal the underlying linear structure of the equation.
Let's apply the distributive property to the left side of the equation: 7(2x+3) + 8(4x-5). We multiply 7 by both 2x and 3, and we multiply 8 by both 4x and -5. This yields the following expansion:
14x + 21 + 32x - 40
Similarly, we apply the distributive property to the right side of the equation: 12(2x+5) + 9. Multiplying 12 by both 2x and 5, we obtain:
24x + 60 + 9
Now, our equation has transformed from its initial complex form to a more manageable state: 14x + 21 + 32x - 40 = 24x + 60 + 9. This simplified equation no longer contains parentheses, making it easier to identify and combine like terms.
4. Step 2: Combine Like Terms for Clarity
With the parentheses successfully eliminated, our next task is to combine like terms on both sides of the equation. Like terms are terms that share the same variable raised to the same power. In our equation, the like terms on the left side are 14x and 32x, and the constant terms are 21 and -40. On the right side, the like terms are 24x and the constant terms are 60 and 9.
Combining the like terms on the left side, we add 14x and 32x to get 46x, and we add 21 and -40 to get -19. This simplifies the left side of the equation to:
46x - 19
Similarly, combining the like terms on the right side, we add 60 and 9 to get 69. This simplifies the right side of the equation to:
24x + 69
Our equation now takes a more streamlined form: 46x - 19 = 24x + 69. By combining like terms, we have reduced the equation to its essential components, making it easier to isolate the variable 'x'.
5. Step 3: Isolate the Variable – The Art of Transposition
Our primary goal is to isolate the variable 'x' on one side of the equation. To achieve this, we employ the technique of transposition, which involves moving terms from one side of the equation to the other while maintaining the equation's balance. This is accomplished by performing the same operation on both sides of the equation, ensuring that the equality remains intact.
To isolate 'x', we begin by moving all terms containing 'x' to one side of the equation and all constant terms to the other side. Let's subtract 24x from both sides of the equation to move the 'x' term from the right side to the left side:
46x - 19 - 24x = 24x + 69 - 24x
This simplifies to:
22x - 19 = 69
Next, we add 19 to both sides of the equation to move the constant term from the left side to the right side:
22x - 19 + 19 = 69 + 19
This simplifies to:
22x = 88
With the variable term isolated on one side and the constant term on the other, we are now just one step away from solving for 'x'.
6. Step 4: Divide and Conquer – The Final Act
With the variable term isolated, the final step in solving for 'x' involves dividing both sides of the equation by the coefficient of 'x'. In our case, the coefficient of 'x' is 22. Dividing both sides of the equation by 22, we get:
22x / 22 = 88 / 22
This simplifies to:
x = 4
Thus, we have successfully solved the equation 7(2x+3)+8(4x-5)=12(2x+5)+9, and the solution is x = 4. This value of 'x' satisfies the original equation, meaning that if we substitute x = 4 back into the equation, both sides will be equal.
7. Verification: Ensuring Accuracy
In the realm of mathematics, precision is paramount. Therefore, it is always prudent to verify our solution to ensure its accuracy. To verify our solution x = 4, we substitute it back into the original equation and check if both sides of the equation are indeed equal.
Substituting x = 4 into the left side of the equation, we get:
7(2(4)+3) + 8(4(4)-5) = 7(8+3) + 8(16-5) = 7(11) + 8(11) = 77 + 88 = 165
Substituting x = 4 into the right side of the equation, we get:
12(2(4)+5) + 9 = 12(8+5) + 9 = 12(13) + 9 = 156 + 9 = 165
Since both sides of the equation evaluate to 165 when x = 4, we have successfully verified our solution. This confirms that x = 4 is indeed the correct solution to the equation.
8. Conclusion: The Power of Algebraic Manipulation
In this comprehensive guide, we have meticulously dissected the process of solving the linear equation 7(2x+3)+8(4x-5)=12(2x+5)+9. We began by establishing a solid understanding of linear equations, then systematically applied the distributive property, combined like terms, isolated the variable, and finally, verified our solution.
Through this step-by-step journey, we have not only arrived at the solution x = 4, but also reinforced the fundamental principles of algebraic manipulation. The ability to solve linear equations is a crucial skill that extends far beyond the confines of the classroom. It is a powerful tool that empowers us to tackle a wide range of real-world problems, from financial calculations to scientific investigations.
By mastering the techniques outlined in this article, you can confidently approach and solve a variety of linear equations, unlocking the power of algebra to unravel the complexities of the world around us.