Solving The Linear Equation 7x - 9 = 28 + 4(x - 1) A Step-by-Step Guide
In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by a variable raised to the power of one, appear frequently in various mathematical contexts and real-world applications. Mastering the techniques to solve these equations is crucial for anyone delving into algebra and beyond. This article provides a comprehensive guide to solving the linear equation 7x - 9 = 28 + 4(x - 1), breaking down each step with detailed explanations. This specific equation serves as an excellent example to illustrate the common methods used in solving linear equations, including the distributive property, combining like terms, and isolating the variable. By understanding the process involved in solving this equation, you will be well-equipped to tackle other similar problems. Whether you're a student learning the basics of algebra or someone looking to refresh your mathematical skills, this guide will provide you with a clear and concise approach to solving linear equations. The beauty of mathematics lies in its logical structure and the step-by-step methods that allow us to find solutions to complex problems. This article embodies that spirit by meticulously explaining each stage in solving the given linear equation, ensuring that you not only understand the answer but also the underlying principles that make it possible. So, let's embark on this mathematical journey and unlock the solution to the equation 7x - 9 = 28 + 4(x - 1).
1. Simplify the Equation Using the Distributive Property
The first step in solving the linear equation 7x - 9 = 28 + 4(x - 1) involves simplifying the equation by applying the distributive property. The distributive property states that a(b + c) = ab + ac. In our equation, we have the term 4(x - 1), which needs to be simplified. This property is a cornerstone of algebraic manipulation, allowing us to remove parentheses and combine like terms. By correctly applying the distributive property, we can transform complex equations into simpler forms that are easier to solve. This step is crucial as it sets the stage for further simplification and isolation of the variable. The distributive property not only simplifies the equation but also makes it more manageable, allowing us to proceed with subsequent steps more effectively. Ignoring this step or misapplying the property can lead to incorrect solutions, so it's essential to understand and execute it accurately. In the context of linear equations, the distributive property often acts as a gateway to unraveling the equation's structure and bringing us closer to finding the value of the variable. Now, let's apply this property to our equation: 4(x - 1) becomes 4 * x - 4 * 1, which simplifies to 4x - 4. Replacing 4(x - 1) with 4x - 4 in the original equation gives us a new, simplified equation that is ready for the next steps in the solution process. This initial simplification is a critical foundation upon which the rest of the solution will be built, ensuring that we are working with the most manageable form of the equation possible.
So, applying the distributive property, we get:
7x - 9 = 28 + 4x - 4
2. Combine Like Terms
After applying the distributive property, the next crucial step in solving the linear equation 7x - 9 = 28 + 4x - 4 is to combine like terms. Like terms are those that have the same variable raised to the same power, or constants. In our equation, we have constant terms on the right side: 28 and -4. Combining like terms simplifies the equation further, making it easier to isolate the variable. This process is a fundamental aspect of algebraic manipulation, allowing us to consolidate terms and reduce the complexity of the equation. The concept of combining like terms is not just limited to this specific equation; it's a universally applicable technique in algebra and beyond. By identifying and combining like terms, we streamline the equation, bringing us closer to the solution. This step ensures that we are working with the most concise form of the equation, which is essential for accurate and efficient problem-solving. Failing to combine like terms correctly can lead to confusion and errors in the subsequent steps. Therefore, mastering this skill is paramount for anyone looking to excel in algebra. In our equation, combining the constants 28 and -4 on the right side simplifies the equation significantly. This consolidation of terms allows us to see the equation more clearly and make strategic decisions about how to proceed with isolating the variable. With the like terms combined, we are one step closer to the final solution, demonstrating the power of algebraic simplification. Let's combine 28 and -4:
28 - 4 = 24
So, the equation becomes:
7x - 9 = 4x + 24
3. Isolate the Variable Term
With the equation simplified to 7x - 9 = 4x + 24, the next pivotal step is to isolate the variable term. Isolating the variable term means getting all the terms containing 'x' on one side of the equation and all the constant terms on the other side. This is a critical technique in solving linear equations, as it brings us closer to determining the value of the variable. The process of isolating the variable involves using inverse operations to move terms across the equality sign. Understanding this process is fundamental to solving not just linear equations but also a wide range of algebraic problems. The goal here is to manipulate the equation in a way that the 'x' term stands alone on one side, making it clear what operations need to be performed next to find the value of 'x'. Failing to isolate the variable term effectively can lead to confusion and make it difficult to solve the equation accurately. Therefore, it's essential to approach this step systematically, ensuring that each operation is performed correctly on both sides of the equation to maintain balance. In our equation, we need to move the '4x' term from the right side to the left side. This is achieved by subtracting '4x' from both sides of the equation. This action cancels out the '4x' term on the right side and leaves us with an equation where the 'x' terms are grouped together on the left side. This strategic move is a key part of the solution process, demonstrating how algebraic manipulations can simplify equations and bring us closer to the answer. By carefully isolating the variable term, we pave the way for the final step of solving for 'x', showcasing the power of algebraic techniques in unraveling mathematical problems. To isolate the variable term, subtract 4x from both sides:
7x - 4x - 9 = 4x - 4x + 24
This simplifies to:
3x - 9 = 24
4. Isolate the Variable
Now that the equation is 3x - 9 = 24, the penultimate step is to isolate the variable 'x' completely. This involves getting 'x' by itself on one side of the equation. This is a critical step in solving any linear equation, as it directly leads to the solution. The process of isolating the variable typically involves performing inverse operations to remove any constants or coefficients that are attached to the variable. Understanding this process is essential for solving a wide range of algebraic problems. The goal here is to manipulate the equation so that 'x' is the only term on one side, allowing us to clearly see its value. Failing to isolate the variable correctly will prevent us from finding the correct solution. Therefore, careful attention to detail and a systematic approach are crucial in this step. In our equation, we first need to remove the constant term '-9' from the left side. This is achieved by adding '9' to both sides of the equation. This action cancels out the '-9' on the left side and moves us closer to having 'x' completely isolated. This strategic move is a key part of the solution process, demonstrating how inverse operations can be used to simplify equations and reveal the value of the variable. Once the constant term is removed, we will be left with a simple equation that can be solved directly for 'x', showcasing the power of algebraic manipulation in solving mathematical problems. To isolate 'x', we first add 9 to both sides:
3x - 9 + 9 = 24 + 9
This simplifies to:
3x = 33
Next, divide both sides by 3:
3x / 3 = 33 / 3
5. Solve for x
The final step in solving the linear equation 3x = 33 is to solve for x. This involves performing the necessary operation to get x by itself on one side of the equation. This is the culmination of all the previous steps and provides the solution to the equation. The process of solving for x typically involves dividing both sides of the equation by the coefficient of x, if there is one. Understanding this process is crucial for solving any linear equation and is a fundamental skill in algebra. The goal here is to determine the numerical value of x that satisfies the original equation. Failing to solve for x correctly will result in an incorrect solution. Therefore, it's essential to approach this step carefully and ensure that the division is performed accurately. In our equation, we need to divide both sides by 3, which is the coefficient of x. This action isolates x on the left side and gives us the numerical value of x on the right side. This strategic move is the final piece of the puzzle, demonstrating how algebraic operations can be used to find the solution to a linear equation. By accurately solving for x, we complete the process of solving the equation and gain a deeper understanding of the principles of algebra. This final step underscores the importance of each preceding step in the solution process, highlighting the logical and systematic nature of mathematical problem-solving. Now, let's divide both sides by 3:
3x / 3 = 33 / 3
This simplifies to:
x = 11
Therefore, the solution to the equation 7x - 9 = 28 + 4(x - 1) is x = 11.
Summary
In summary, solving the linear equation 7x - 9 = 28 + 4(x - 1) involves a series of steps that utilize fundamental algebraic principles. The first step is to simplify the equation using the distributive property, which involves multiplying the term outside the parentheses by each term inside the parentheses. This step expands the equation and prepares it for further simplification. The second step is to combine like terms, which involves adding or subtracting terms that have the same variable and exponent. This step reduces the number of terms in the equation and makes it easier to work with. The third step is to isolate the variable term, which involves moving all terms containing the variable to one side of the equation and all constant terms to the other side. This step sets the stage for isolating the variable itself. The fourth step is to isolate the variable, which involves performing inverse operations to undo any operations that are being performed on the variable. This step gets the variable by itself on one side of the equation. The final step is to solve for x, which involves dividing both sides of the equation by the coefficient of x. This step gives the value of x that satisfies the equation. These steps are not only applicable to this specific equation but also form a general strategy for solving linear equations. By mastering these steps, one can confidently approach and solve a wide range of linear equations. The process of solving linear equations is a cornerstone of algebra and is essential for further studies in mathematics and related fields. The equation we solved, 7x - 9 = 28 + 4(x - 1), is a classic example of a linear equation, and the steps we followed demonstrate the power and elegance of algebraic techniques. Understanding these techniques not only allows us to solve equations but also enhances our problem-solving skills in other areas of mathematics and life. So, the solution x = 11 is not just an answer; it's a testament to the logical and systematic approach that makes mathematics such a powerful tool.