Solving The Equation 2x + 3(3x - 5) = 51 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. This article will delve into the process of solving a specific linear equation, namely, $2x + 3(3x - 5) = 51$. We will break down each step, ensuring a clear understanding of how to arrive at the solution. This detailed explanation is crucial for anyone looking to strengthen their algebra skills, whether you're a student tackling homework or simply someone interested in revisiting mathematical concepts. So, let’s embark on this mathematical journey together and unravel the value of $x$ that satisfies this equation.

Understanding the Basics of Linear Equations

Before we jump into solving the equation, let’s establish a firm grasp on what linear equations are and the principles that govern their solutions. 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 produce a straight line. The equation we're tackling, $2x + 3(3x - 5) = 51$, fits this description perfectly. It involves a single variable, $x$, and no exponents or other complex functions. The highest power of the variable in a linear equation is always 1.

The core principle in solving linear equations is maintaining balance. Think of an equation as a perfectly balanced scale. Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the scale balanced. This ensures that the equality remains true throughout the solving process. Common operations include addition, subtraction, multiplication, and division. The goal is to isolate the variable on one side of the equation, revealing its value. In our case, we want to isolate $x$ on one side, effectively finding the numerical value that makes the equation true.

Understanding these basics is paramount to successfully solving not just this equation, but any linear equation you encounter. It lays the foundation for more advanced algebraic concepts and problem-solving techniques. With a solid grasp of these principles, we can confidently move forward and solve our equation step-by-step.

Step-by-Step Solution of the Equation

Now, let's meticulously solve the equation $2x + 3(3x - 5) = 51$. We'll break down each step, explaining the rationale behind it to ensure clarity.

Step 1: Distribute the Constant

The first step in solving this equation is to simplify it by distributing the constant outside the parentheses. In our equation, we have $3(3x - 5)$. This means we need to multiply both terms inside the parentheses by 3. Applying the distributive property, we get:

$3 * 3x = 9x$

$3 * -5 = -15$

So, $3(3x - 5)$ becomes $9x - 15$. Our equation now looks like this:

$2x + 9x - 15 = 51$

This step is crucial because it eliminates the parentheses, making the equation easier to manipulate. Distribution is a fundamental operation in algebra, and mastering it is key to solving more complex equations.

Step 2: Combine Like Terms

Next, we need to simplify the equation further by combining like terms. Like terms are terms that have the same variable raised to the same power. In our equation, we have two terms with $x$: $2x$ and $9x$. We can combine these by adding their coefficients:

$2x + 9x = 11x$

Our equation now simplifies to:

$11x - 15 = 51$

Combining like terms is another essential step in simplifying equations. It reduces the number of terms, making the equation more manageable and bringing us closer to isolating the variable.

Step 3: Isolate the Variable Term

Our next goal is to isolate the term containing the variable, which in this case is $11x$. To do this, we need to get rid of the constant term on the same side of the equation, which is $-15$. We can do this by adding 15 to both sides of the equation:

$11x - 15 + 15 = 51 + 15$

This simplifies to:

$11x = 66$

Remember, the principle of maintaining balance is crucial here. By adding 15 to both sides, we ensure that the equation remains equal. We have now successfully isolated the variable term on one side of the equation.

Step 4: Solve for x

Finally, we need to solve for $x$ by isolating it completely. Currently, $x$ is being multiplied by 11. To undo this multiplication, we need to divide both sides of the equation by 11:

$11x / 11 = 66 / 11$

This simplifies to:

$x = 6$

And there we have it! We have successfully solved the equation and found the value of $x$. This final step demonstrates the inverse relationship between multiplication and division and how we can use it to isolate the variable.

The Solution: x = 6

After meticulously working through each step, we've arrived at the solution to the equation $2x + 3(3x - 5) = 51$. The value of $x$ that satisfies this equation is 6. This means that if we substitute 6 for $x$ in the original equation, both sides will be equal. To be absolutely sure of our answer, let's perform a quick check.

Verifying the Solution

To verify our solution, we substitute $x = 6$ back into the original equation:

$2(6) + 3(3(6) - 5) = 51$

Now, we simplify step-by-step:

$2(6) = 12$

$3(6) = 18$

$18 - 5 = 13$

$3(13) = 39$

So, the left side of the equation becomes:

$12 + 39 = 51$

This matches the right side of the equation, which is 51. Therefore, our solution $x = 6$ is correct. Verification is an important step in problem-solving. It provides assurance that the solution we've found is accurate and that we haven't made any errors along the way.

Importance of Understanding the Order of Operations

Throughout the process of solving this equation, we implicitly followed the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and adhering to the order of operations is paramount in solving any mathematical equation. Let's briefly discuss why it's so crucial.

The order of operations dictates the sequence in which we perform mathematical operations. If we deviate from this order, we risk arriving at an incorrect solution. For instance, in our equation, we first dealt with the parentheses by distributing the constant, then we combined like terms, and finally, we isolated the variable. If we had, say, added 2x and 3 before distributing the 3, we would have introduced an error. PEMDAS ensures consistency and clarity in mathematical calculations.

Parentheses are always addressed first, as they group terms together. Exponents come next, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). This hierarchy ensures that complex expressions are evaluated unambiguously. In the context of our equation, understanding PEMDAS allowed us to correctly simplify the expression and arrive at the accurate value of $x$. Mastering the order of operations is not just important for solving equations; it's a fundamental skill that underpins all areas of mathematics.

Real-World Applications of Linear Equations

Linear equations aren't just abstract mathematical concepts; they have a plethora of real-world applications. Understanding how to solve them can be incredibly useful in various everyday scenarios. Let's explore a few examples to illustrate the practical significance of linear equations.

One common application is in calculating costs and expenses. For example, imagine you're planning a road trip and need to budget for gasoline. If you know the price per gallon, your car's fuel efficiency (miles per gallon), and the total distance you'll be traveling, you can set up a linear equation to determine the total cost of gasoline. Similarly, linear equations can be used to calculate the total cost of a service, like a plumber's visit, where there's a fixed call-out fee plus an hourly rate.

Another application lies in financial calculations. Linear equations are used to determine simple interest on loans or investments. The total amount of money you'll have after a certain period can be expressed as a linear equation, taking into account the principal amount, interest rate, and time. Budgeting and financial planning often involve using linear equations to project income, expenses, and savings.

Linear equations also play a role in physics and engineering. For instance, Ohm's Law, which relates voltage, current, and resistance in an electrical circuit, is a linear equation. Engineers use this law to design and analyze circuits. In physics, linear equations can describe the motion of objects at constant velocity. The distance traveled is equal to the product of velocity and time, a classic linear relationship.

The equation we solved, $2x + 3(3x - 5) = 51$, might seem purely academic, but the underlying principles of solving linear equations are applicable in countless real-world situations. By mastering these skills, you're equipping yourself with a powerful tool for problem-solving in various domains.

Conclusion

In this comprehensive guide, we've successfully solved the equation $2x + 3(3x - 5) = 51$, demonstrating a step-by-step approach to linear equation solving. We began by understanding the basics of linear equations and the principle of maintaining balance. We then meticulously worked through each step: distributing the constant, combining like terms, isolating the variable term, and finally, solving for $x$. We arrived at the solution $x = 6$ and verified its correctness by substituting it back into the original equation. We also emphasized the importance of the order of operations (PEMDAS) and explored the real-world applications of linear equations.

Solving equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts and problem-solving techniques. The process we've outlined here can be applied to a wide range of linear equations. The key is to understand the underlying principles and to practice consistently. By breaking down complex problems into smaller, manageable steps, you can confidently tackle any equation that comes your way. Remember, mathematics is not just about memorizing formulas; it's about developing a logical and systematic approach to problem-solving. We encourage you to continue practicing and exploring the fascinating world of mathematics.