Solve The Equation 6(3x - 15) = 36
In this article, we will delve into the process of solving the algebraic equation 6(3x - 15) = 36. This equation falls under the category of linear equations, which are fundamental in algebra. Understanding how to solve these equations is crucial for more advanced mathematical concepts. We will break down the solution into manageable steps, ensuring a clear and comprehensive understanding of the underlying principles. The focus will be on applying the distributive property, combining like terms, and isolating the variable x to find its value. This step-by-step approach will not only provide the solution but also enhance your problem-solving skills in algebra. Whether you're a student learning algebra for the first time or someone looking to refresh your knowledge, this guide will offer a solid foundation and a practical method for solving similar equations. Let's embark on this mathematical journey together, unraveling the complexities of this equation and discovering the value of x.
Step 1: Apply the Distributive Property
The first crucial step in solving the equation 6(3x - 15) = 36 involves applying the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. In this case, we need to distribute the 6 across the terms (3x) and (-15). This means we will multiply 6 by both 3x and -15 separately. This process is essential for simplifying the equation and making it easier to solve. By correctly applying the distributive property, we eliminate the parentheses and pave the way for further simplification. It’s a cornerstone of algebraic manipulation, allowing us to transform complex expressions into more manageable forms. Misapplication of this property can lead to incorrect solutions, so it's vital to grasp this concept firmly. Let's proceed with the multiplication to see how it unfolds and simplifies our equation.
When we multiply 6 by 3x, we get 18x. Next, we multiply 6 by -15, which results in -90. Therefore, applying the distributive property to the left side of the equation transforms 6(3x - 15) into 18x - 90. The equation now looks like this: 18x - 90 = 36. This transformation is a significant step forward, as we have successfully removed the parentheses, making the equation much simpler to work with. The next steps will involve isolating the term with x and then solving for x. This initial application of the distributive property is often the key to unlocking the solution in many algebraic problems. Understanding and mastering this step is fundamental for success in algebra and beyond.
Step 2: Isolate the Term with 'x'
Having applied the distributive property, our equation now stands as 18x - 90 = 36. The next critical step is to isolate the term containing the variable x, which in this case is 18x. To achieve this, we need to eliminate the constant term that is on the same side of the equation. In our current equation, the constant term is -90. The fundamental principle we employ here is the addition property of equality. This property states that adding the same value to both sides of an equation maintains the equality. Therefore, to eliminate -90, we need to add its opposite, which is +90, to both sides of the equation. This ensures that the equation remains balanced and that we are performing a valid algebraic manipulation. Isolating the term with x is a crucial step because it brings us closer to determining the value of x. Let's proceed with adding 90 to both sides of the equation and observe the transformation.
Adding 90 to both sides of the equation 18x - 90 = 36 gives us: 18x - 90 + 90 = 36 + 90. On the left side, -90 and +90 cancel each other out, leaving us with just 18x. On the right side, 36 + 90 equals 126. So, the equation simplifies to 18x = 126. This is a significant simplification, as we have successfully isolated the term with x on one side of the equation. Now, the equation is in a form where we can easily solve for x by performing the inverse operation of multiplication, which is division. This step highlights the importance of using inverse operations to isolate variables, a fundamental technique in solving algebraic equations. The next step will involve dividing both sides of the equation by the coefficient of x to finally determine the value of x.
Step 3: Solve for 'x'
Now that we have isolated the term with x, our equation is 18x = 126. The final step in solving for x involves getting x by itself on one side of the equation. Currently, x is being multiplied by 18. To undo this multiplication and isolate x, we need to perform the inverse operation, which is division. The division property of equality dictates that dividing both sides of an equation by the same non-zero number maintains the equality. Therefore, we will divide both sides of the equation by 18. This will effectively cancel out the 18 on the left side, leaving x alone. This step is the culmination of our efforts, as it directly leads to the solution for x. Let's proceed with the division and see what the value of x is.
Dividing both sides of the equation 18x = 126 by 18 gives us: 18x / 18 = 126 / 18. On the left side, 18x divided by 18 simplifies to x. On the right side, 126 divided by 18 equals 7. Therefore, the equation simplifies to x = 7. This is our final solution. We have successfully solved the equation 6(3x - 15) = 36 and found that x equals 7. This process illustrates the power of algebraic manipulation in solving equations. By applying the distributive property, isolating the term with x, and using inverse operations, we were able to systematically arrive at the solution. This method can be applied to a wide range of linear equations, making it a valuable skill in mathematics.
Step 4: Verification of the Solution
After solving an equation, it is always a good practice to verify the solution. This ensures that the value we have found for the variable x is indeed correct. To verify our solution x = 7 for the equation 6(3x - 15) = 36, we substitute 7 in place of x in the original equation. This process of substitution allows us to check if both sides of the equation are equal, confirming the validity of our solution. Verification is a crucial step as it helps to catch any errors that may have occurred during the solving process. It also reinforces the understanding of the equation and the solution. Let's proceed with the substitution and see if our solution holds true.
Substituting x with 7 in the original equation 6(3x - 15) = 36, we get: 6(3(7) - 15) = 36. First, we simplify the expression inside the parentheses. 3 multiplied by 7 is 21, so we have: 6(21 - 15) = 36. Next, we subtract 15 from 21, which gives us 6: 6(6) = 36. Finally, we multiply 6 by 6, which equals 36. So, we have: 36 = 36. Since both sides of the equation are equal, our solution x = 7 is indeed correct. This verification step provides confidence in our solution and demonstrates the accuracy of our step-by-step approach. It also highlights the importance of checking one's work in mathematics to ensure correctness. This comprehensive process, from applying algebraic principles to verifying the solution, is essential for mastering equation-solving skills.
Conclusion
In conclusion, we have successfully solved the equation 6(3x - 15) = 36 by following a systematic, step-by-step approach. We began by applying the distributive property, which allowed us to simplify the equation by removing the parentheses. This involved multiplying the term outside the parentheses (6) by each term inside (3x and -15). Next, we isolated the term containing the variable x by adding 90 to both sides of the equation. This step is crucial in bringing us closer to solving for x. Then, we solved for x by dividing both sides of the equation by 18, the coefficient of x. This gave us the solution x = 7. Finally, we verified our solution by substituting 7 back into the original equation, confirming its correctness. This process underscores the importance of understanding and applying fundamental algebraic principles such as the distributive property and the properties of equality. Each step is a building block towards the final solution, and mastering these steps is essential for success in algebra and beyond. The ability to solve equations like this is a foundational skill that is applicable in various fields, from science and engineering to finance and everyday problem-solving. By understanding the underlying concepts and practicing regularly, anyone can develop proficiency in solving algebraic equations.