Solving 7x - 3(x - 6) = 30 A Step-by-Step Guide

This article delves into the step-by-step solution of the equation 7x - 3(x - 6) = 30. Understanding how to solve linear equations is a fundamental skill in algebra, and this example provides a clear and concise illustration of the process. We will break down each step, explaining the underlying principles and techniques involved. By the end of this guide, you will not only know the correct answer but also understand the reasoning behind it. So, let's dive in and unravel this equation together!

Understanding the Basics of Linear Equations

Before we tackle the specific equation, it's essential to grasp the basics 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, when graphed, they form a straight line. The goal in solving a linear equation is to isolate the variable (in this case, 'x') on one side of the equation, thereby determining its value. This isolation is achieved by performing the same operations on both sides of the equation, maintaining the balance and equality.

The fundamental principle behind solving any equation is the principle of equality. This principle states that you can add, subtract, multiply, or divide both sides of an equation by the same number without changing the equation's solution. This is the cornerstone of algebraic manipulation. We use this principle repeatedly throughout the process of solving the equation. For example, if we have the equation a = b, then a + c = b + c, a - c = b - c, a * c = b * c, and a / c = b / c (where c ≠ 0) are all valid transformations.

Another crucial concept is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations should be performed to correctly evaluate an expression. In our equation, we will first deal with the parentheses, then perform any multiplication or division, and finally address addition and subtraction. Adhering to the order of operations ensures that we simplify the equation correctly and arrive at the accurate solution.

Step 1: Distribute the -3

The first step in solving the equation 7x - 3(x - 6) = 30 is to eliminate the parentheses. This is achieved by distributing the -3 across the terms inside the parentheses. Remember that distributing means multiplying the term outside the parentheses by each term inside the parentheses. So, we multiply -3 by both 'x' and '-6'.

When we multiply -3 by 'x', we get -3x. When we multiply -3 by -6, we get +18 (since a negative times a negative is a positive). Therefore, the equation now becomes:

7x - 3x + 18 = 30

This step is crucial because it simplifies the equation by removing the parentheses, making it easier to combine like terms. The distributive property is a fundamental concept in algebra, and mastering it is essential for solving more complex equations. It allows us to break down expressions and manipulate them in a way that reveals the underlying relationships between variables and constants. By correctly distributing the -3, we have transformed the equation into a more manageable form, setting the stage for the next steps in the solution process.

Step 2: Combine Like Terms

After distributing the -3, our equation looks like this: 7x - 3x + 18 = 30. The next step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In this equation, 7x and -3x are like terms because they both contain the variable 'x' raised to the power of 1. We can combine these terms by simply adding or subtracting their coefficients (the numbers in front of the variable).

To combine 7x and -3x, we subtract the coefficients: 7 - 3 = 4. So, 7x - 3x simplifies to 4x. The equation now becomes:

4x + 18 = 30

Combining like terms is a fundamental step in simplifying algebraic expressions and equations. It allows us to consolidate terms and make the equation more concise. This step makes it easier to isolate the variable and solve for its value. By combining the 'x' terms, we have reduced the complexity of the equation and moved closer to our goal of isolating 'x'. This process of simplification is a key strategy in algebra, allowing us to break down complex problems into smaller, more manageable steps.

Step 3: Isolate the Variable Term

Now that we have simplified the equation to 4x + 18 = 30, our next goal is to isolate the variable term, which is 4x. To do this, we need to eliminate the constant term, which is +18, from the left side of the equation. We can achieve this by using the principle of equality, which states that we can perform the same operation on both sides of the equation without changing its solution. In this case, we will subtract 18 from both sides of the equation.

Subtracting 18 from both sides, we get:

4x + 18 - 18 = 30 - 18

This simplifies to:

4x = 12

By subtracting 18 from both sides, we have successfully isolated the variable term (4x) on the left side of the equation. This step is crucial because it brings us closer to solving for the value of 'x'. The principle of equality is the foundation of this step, ensuring that we maintain the balance of the equation while manipulating it to isolate the variable. Isolating the variable term is a common strategy in solving algebraic equations, and it is essential for determining the unknown value.

Step 4: Solve for x

We've reached the final step in solving the equation. Our equation is now 4x = 12. To solve for 'x', we need to isolate 'x' completely. This means we need to eliminate the coefficient 4 from the left side of the equation. Again, we will use the principle of equality. Since 'x' is being multiplied by 4, we will divide both sides of the equation by 4.

Dividing both sides by 4, we get:

4x / 4 = 12 / 4

This simplifies to:

x = 3

Therefore, the solution to the equation 7x - 3(x - 6) = 30 is x = 3. We have successfully isolated 'x' and determined its value. This final step demonstrates the power of algebraic manipulation and the principle of equality. By dividing both sides by the coefficient of 'x', we have effectively undone the multiplication and revealed the value of the variable. This process is a fundamental technique in algebra and is used extensively in solving various types of equations.

Verifying the Solution

To ensure that our solution is correct, it's always a good practice to verify it. Verification involves substituting the value we found for 'x' back into the original equation and checking if both sides of the equation are equal. This process helps us catch any errors we might have made during the solution process and gives us confidence in our answer.

Our original equation was: 7x - 3(x - 6) = 30. We found that x = 3. Let's substitute x = 3 into the equation:

7(3) - 3(3 - 6) = 30

Now, we simplify both sides of the equation following the order of operations:

21 - 3(-3) = 30

21 + 9 = 30

30 = 30

Since both sides of the equation are equal, our solution x = 3 is correct. Verification is a crucial step in problem-solving, especially in mathematics. It provides a check and balance system, ensuring that our solution is accurate and reliable. By substituting the value back into the original equation, we can confirm that our algebraic manipulations were correct and that we have indeed found the solution that satisfies the equation.

Conclusion

In conclusion, we have successfully solved the equation 7x - 3(x - 6) = 30 and found that x = 3. We achieved this by following a step-by-step process that included distributing, combining like terms, isolating the variable term, and finally solving for 'x'. We also verified our solution by substituting it back into the original equation. This example demonstrates the importance of understanding the fundamental principles of algebra, such as the distributive property, combining like terms, and the principle of equality. Mastering these concepts is crucial for solving a wide range of algebraic equations and problems.

The process of solving equations is not just about finding the right answer; it's also about developing problem-solving skills and logical reasoning. Each step we took in solving this equation involved a specific strategy and a clear understanding of the underlying principles. By breaking down the problem into smaller, manageable steps, we were able to systematically arrive at the solution. This approach can be applied to various other mathematical problems and even to real-life situations. Remember, practice is key to mastering algebra and problem-solving. The more equations you solve, the more comfortable and confident you will become in your abilities.

Therefore, the correct answer to the question "What is the solution to this equation? 7x - 3(x - 6) = 30" is D. x = 3.