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 their straight-line graphical representation, appear in various fields, from basic algebra to advanced calculus and beyond. This guide provides a comprehensive approach to solving the linear equation 7x - 9 = 28 + 4(x - 1), walking you through each step with clarity and precision. Understanding the methodology behind solving such equations is crucial for building a strong foundation in mathematical problem-solving.
Understanding Linear Equations
Before we dive into the specifics of solving 7x - 9 = 28 + 4(x - 1), it's important to grasp the essence of 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. These equations are called "linear" because they describe a straight line when plotted on a graph. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x represents the variable we aim to solve for. The goal is to isolate the variable on one side of the equation to determine its value.
Linear equations are ubiquitous in mathematics and its applications. They model relationships between quantities that change at a constant rate, making them essential tools in physics, engineering, economics, and computer science. From calculating the trajectory of a projectile to determining the break-even point in a business venture, linear equations provide a framework for understanding and solving real-world problems. The process of solving linear equations involves applying algebraic principles to manipulate the equation while maintaining its balance, ultimately revealing the value of the unknown variable.
In the context of 7x - 9 = 28 + 4(x - 1), we encounter a linear equation with parentheses, which adds a layer of complexity. However, by systematically applying the distributive property and combining like terms, we can reduce the equation to a simpler form and isolate the variable x. The following steps will guide you through the process, ensuring you not only arrive at the correct solution but also understand the underlying principles that govern linear equations.
Step-by-Step Solution
Let's embark on a detailed journey to solve the linear equation 7x - 9 = 28 + 4(x - 1). This process involves several key steps, each built upon fundamental algebraic principles. By following these steps meticulously, you'll gain a deep understanding of how to manipulate equations and isolate variables, a skill that extends far beyond this specific problem. Our goal is not just to find the value of x but to master the technique of solving linear equations in general.
1. Distribute
The first step in solving 7x - 9 = 28 + 4(x - 1) is to eliminate the parentheses. This is achieved by applying the distributive property, which states that a(b + c) = ab + ac. In our case, we need to distribute the 4 across the terms inside the parentheses: 4(x - 1). This means multiplying 4 by both x and -1. The result is 4x - 4. So, after applying the distributive property, our equation transforms from 7x - 9 = 28 + 4(x - 1) to 7x - 9 = 28 + 4x - 4. This step is crucial because it simplifies the equation, making it easier to combine like terms and isolate the variable.
The distributive property is a cornerstone of algebra, allowing us to handle expressions enclosed within parentheses. It's essential to apply it correctly, ensuring that each term inside the parentheses is multiplied by the factor outside. In more complex equations, this might involve distributing multiple terms or even distributing across multiple sets of parentheses. However, the underlying principle remains the same: to expand the expression and eliminate the parentheses. By mastering this step, you'll be well-equipped to tackle a wide range of algebraic problems.
2. Combine Like Terms
Now that we've eliminated the parentheses, the next step is to combine like terms on each side of the equation. In the equation 7x - 9 = 28 + 4x - 4, we have constant terms (-9, 28, and -4) and terms with the variable x (7x and 4x). Like terms are terms that have the same variable raised to the same power (or are constants). We can combine like terms by adding or subtracting their coefficients. On the left side of the equation, there are no like terms to combine. On the right side, we have two constant terms: 28 and -4. Combining these gives us 28 - 4 = 24. Therefore, the equation simplifies to 7x - 9 = 4x + 24. This step is essential for further simplifying the equation and isolating the variable.
Combining like terms is a fundamental technique in algebra that helps to consolidate and simplify expressions. It's based on the principle that we can only add or subtract terms that are "alike." For example, we can combine 3x and 5x because they both involve the variable x raised to the power of 1. Similarly, we can combine the constants 7 and -2. However, we cannot combine 3x and 7 because they are not like terms (one has a variable, and the other is a constant). By carefully combining like terms, we reduce the complexity of the equation, making it easier to manipulate and solve. This skill is crucial for solving not just linear equations but also more complex algebraic problems.
3. Isolate the Variable Term
To isolate the variable term in the equation 7x - 9 = 4x + 24, we need to move all terms containing x to one side of the equation. A common approach is to subtract the smaller x term from both sides. In this case, we have 7x on the left and 4x on the right. Subtracting 4x from both sides maintains the balance of the equation while moving the x term to the left side. This gives us 7x - 4x - 9 = 4x - 4x + 24, which simplifies to 3x - 9 = 24. This step is crucial because it groups all terms involving the variable on one side, paving the way for isolating the variable itself.
The principle of maintaining balance in an equation is paramount in algebra. Any operation we perform on one side of the equation must also be performed on the other side to ensure that the equality holds true. Subtracting the same term from both sides is one such operation. This technique is not limited to isolating variable terms; it can also be used to isolate constant terms or to simplify equations in various other ways. The key is to choose operations that move us closer to our goal of isolating the variable without disrupting the equation's balance. Mastering this technique is essential for solving a wide range of algebraic equations.
4. Isolate the Variable
Now that we have the equation 3x - 9 = 24, our goal is to isolate x completely. To do this, we first need to eliminate the constant term (-9) on the left side. We can achieve this by adding 9 to both sides of the equation. This gives us 3x - 9 + 9 = 24 + 9, which simplifies to 3x = 33. Adding the same constant to both sides maintains the balance of the equation while moving us closer to isolating x. This step is a crucial application of the addition property of equality, a fundamental principle in algebra.
The addition property of equality states that adding the same value to both sides of an equation does not change the equation's solution. This property, along with the subtraction property of equality, allows us to manipulate equations to isolate variables. In the context of 3x - 9 = 24, adding 9 to both sides cancels out the -9 on the left, leaving us with just the term involving x. This is a common strategy in solving equations: to use inverse operations (addition and subtraction, multiplication and division) to "undo" the operations that are preventing us from isolating the variable. Understanding and applying these properties is essential for solving algebraic equations effectively.
5. Solve for x
We've reached the final step in solving for x in the equation 3x = 33. To isolate x, we need to undo the multiplication by 3. The inverse operation of multiplication is division, so we will divide both sides of the equation by 3. This gives us 3x / 3 = 33 / 3, which simplifies to x = 11. This is the solution to the equation 7x - 9 = 28 + 4(x - 1). Dividing both sides by the same non-zero number is a valid operation because it maintains the balance of the equation, a fundamental principle in algebra.
The division property of equality, like the addition and subtraction properties, is a cornerstone of equation solving. It states that dividing both sides of an equation by the same non-zero value does not change the equation's solution. This property allows us to isolate variables that are multiplied by a constant. In the case of 3x = 33, dividing by 3 effectively "undoes" the multiplication, leaving us with x on one side and its value on the other. This step, combined with the previous steps, demonstrates the power of using inverse operations to solve equations. By mastering these techniques, you'll be able to tackle a wide range of algebraic problems with confidence.
Checking the Solution
To ensure the accuracy of our solution, it's always a good practice to check our answer. We found that x = 11 is the solution to the equation 7x - 9 = 28 + 4(x - 1). To check this, we substitute x = 11 back into the original equation and verify that both sides are equal. This process is called "plugging in" the solution. It's a crucial step in problem-solving because it helps us catch any errors we might have made during the solution process. A correct solution will make the equation a true statement, while an incorrect solution will not.
Substituting x = 11 into the left side of the equation 7x - 9 gives us 7(11) - 9, which is 77 - 9 = 68. Now, we substitute x = 11 into the right side of the equation 28 + 4(x - 1), which gives us 28 + 4(11 - 1). This simplifies to 28 + 4(10) = 28 + 40 = 68. Since both sides of the equation evaluate to 68 when x = 11, we can confidently say that our solution is correct. This verification step reinforces our understanding of the solution process and provides assurance that we have solved the equation accurately.
Conclusion
In conclusion, we have successfully solved the linear equation 7x - 9 = 28 + 4(x - 1) by meticulously following a step-by-step approach. We began by applying the distributive property to eliminate parentheses, then combined like terms to simplify the equation. Next, we isolated the variable term by strategically adding or subtracting terms from both sides. We then isolated the variable itself by using inverse operations, ultimately arriving at the solution x = 11. Finally, we verified our solution by substituting it back into the original equation, confirming its accuracy. This comprehensive process demonstrates the power of algebraic manipulation and the importance of maintaining balance in equations.
Solving linear equations is a fundamental skill in mathematics, and the techniques we've employed here are applicable to a wide range of problems. By mastering these steps, you'll be well-equipped to tackle more complex algebraic challenges. Remember, the key to success in mathematics is not just memorizing formulas but understanding the underlying principles and applying them systematically. The journey through this equation has not only provided us with a solution but also reinforced our understanding of the core concepts of algebra. Keep practicing, and you'll continue to grow your mathematical abilities.
