Solving Linear Equations A Step-by-Step Guide To 4(3x - 6) + 2(3x - 1) = 2

Mastering the art of solving linear equations is a fundamental skill in mathematics, serving as a cornerstone for more advanced concepts. Linear equations, characterized by a variable raised to the first power, appear frequently in various mathematical and real-world scenarios. This article provides a comprehensive guide to solving the linear equation 4(3x - 6) + 2(3x - 1) = 2, breaking down each step with detailed explanations to ensure clarity and understanding. By following this methodical approach, you'll gain the confidence to tackle a wide range of linear equations.

Understanding the Basics of Linear Equations

Before diving into the solution, it's crucial to grasp the fundamental principles of linear equations. A linear equation is an algebraic equation where the highest power of the variable is 1. These equations can be written in the general form ax + b = c, where a, b, and c are constants, and x represents the variable. The goal of solving a linear equation is to isolate the variable on one side of the equation, thereby determining its value. To achieve this, we employ a series of algebraic manipulations based on the properties of equality.

The properties of equality state that performing the same operation on both sides of an equation maintains the equality. These operations include addition, subtraction, multiplication, and division. For instance, if we have an equation a = b, we can add c to both sides (a + c = b + c), subtract c from both sides (a - c = b - c), multiply both sides by c (ac = bc), or divide both sides by c (assuming c is not zero) (a/c = b/c). These properties are the bedrock of solving equations, allowing us to manipulate the equation while preserving its balance. With a solid understanding of these principles, we're well-equipped to embark on solving the equation at hand.

Step 1: Applying the Distributive Property

The first critical step in simplifying the equation 4(3x - 6) + 2(3x - 1) = 2 involves applying the distributive property. The distributive property states that a(b + c) = ab + ac. This property allows us to eliminate the parentheses by multiplying the term outside the parentheses by each term inside. In our equation, we have two instances where the distributive property applies: 4(3x - 6) and 2(3x - 1). Let's apply the distributive property to each of these terms individually.

For the first term, 4(3x - 6), we multiply 4 by both 3x and -6. This gives us 4 * 3x - 4 * 6, which simplifies to 12x - 24. Similarly, for the second term, 2(3x - 1), we multiply 2 by both 3x and -1. This results in 2 * 3x - 2 * 1, which simplifies to 6x - 2. Now, we can rewrite the original equation by replacing the parenthetical expressions with their simplified forms. Our equation becomes 12x - 24 + 6x - 2 = 2. By applying the distributive property, we've successfully removed the parentheses, making the equation easier to manipulate and paving the way for the next step in the solution process.

Step 2: Combining Like Terms

After applying the distributive property, the equation 12x - 24 + 6x - 2 = 2 presents an opportunity to simplify further by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, we have two terms containing the variable x (12x and 6x) and two constant terms (-24 and -2). Combining like terms involves adding or subtracting their coefficients. Let's begin by identifying the like terms in the equation. We have 12x and 6x as the terms containing x, and -24 and -2 as the constant terms.

To combine the x terms, we add their coefficients: 12 + 6 = 18. Therefore, 12x + 6x simplifies to 18x. Next, we combine the constant terms by adding them: -24 + (-2) = -26. So, the constant terms simplify to -26. Now, we can rewrite the equation with the combined like terms. Our equation becomes 18x - 26 = 2. By combining like terms, we've streamlined the equation, making it more concise and easier to solve. This simplification brings us closer to isolating the variable x and finding its value. With the equation now in a more manageable form, we can proceed to the next step in the solution process.

Step 3: Isolating the Variable Term

The next crucial step in solving the linear equation 18x - 26 = 2 is to isolate the variable term, which in this case is 18x. Isolating the variable term means getting it alone on one side of the equation. To achieve this, we need to eliminate the constant term, -26, from the left side of the equation. We can accomplish this by using the addition property of equality. The addition property of equality states that adding the same value to both sides of an equation maintains the equality. Therefore, we can add 26 to both sides of the equation to cancel out the -26 on the left side.

Adding 26 to both sides of the equation 18x - 26 = 2 gives us 18x - 26 + 26 = 2 + 26. On the left side, -26 and +26 cancel each other out, leaving us with 18x. On the right side, 2 + 26 equals 28. Thus, our equation becomes 18x = 28. By adding 26 to both sides, we've successfully isolated the variable term, 18x, on the left side of the equation. This brings us significantly closer to solving for x. Now that the variable term is isolated, we can move on to the final step, which involves solving for the variable itself.

Step 4: Solving for the Variable

With the variable term isolated, the equation is now in the form 18x = 28. To solve for x, we need to isolate x completely. This means getting x by itself on one side of the equation. Since x is being multiplied by 18, we can use the division property of equality to undo this multiplication. The division property of equality states that dividing both sides of an equation by the same non-zero value maintains the equality. Therefore, we can divide both sides of the equation by 18 to isolate x.

Dividing both sides of the equation 18x = 28 by 18 gives us (18x) / 18 = 28 / 18. On the left side, 18 in the numerator and 18 in the denominator cancel each other out, leaving us with x. On the right side, we have the fraction 28 / 18. To simplify this fraction, we can find the greatest common divisor (GCD) of 28 and 18, which is 2. Dividing both the numerator and denominator by 2, we get 14 / 9. Thus, our solution is x = 14 / 9. By dividing both sides of the equation by 18, we've successfully solved for x, finding its value to be 14 / 9. This completes the solution process for the linear equation 4(3x - 6) + 2(3x - 1) = 2. We have systematically applied the distributive property, combined like terms, and used the properties of equality to isolate the variable and determine its value.

Conclusion

In summary, solving the linear equation 4(3x - 6) + 2(3x - 1) = 2 involves a series of well-defined steps: applying the distributive property, combining like terms, isolating the variable term, and finally, solving for the variable. By mastering these steps and understanding the underlying principles of linear equations, you can confidently tackle a wide range of algebraic problems. The solution to the equation is x = 14 / 9. Remember, practice is key to solidifying your understanding and enhancing your problem-solving skills in mathematics. Embrace the challenge of solving linear equations, and you'll find yourself well-equipped to tackle more complex mathematical concepts in the future.