Solve 8 = 3/2(7x - 2) For X

Solving the Equation: A Step-by-Step Guide to Finding 'x'

Welcome, math enthusiasts! Today, we're diving into a common algebraic challenge: solving an equation for an unknown variable. Our focus will be on the specific equation $8 = \frac{3}{2}(7x - 2)$. You might look at this and think, "Where do I even begin?" Don't worry, we'll break it down into simple, manageable steps. The goal here is to isolate 'x', meaning we want to get 'x' all by itself on one side of the equation. This process involves using inverse operations – essentially, doing the opposite of what's being done to 'x' to undo it. We'll tackle this by first dealing with the fraction and then the terms within the parentheses. This method is fundamental in algebra and will serve you well as you encounter more complex problems. Remember, patience and a systematic approach are your best friends when it comes to algebra. So, let's roll up our sleeves and conquer this equation together, transforming a seemingly intimidating problem into a clear path towards a solution. We'll explore the properties of equality, like the multiplication property and the addition property, which allow us to maintain the balance of the equation as we manipulate it. This journey into solving for 'x' is not just about finding a numerical answer; it's about understanding the logic and structure of mathematical expressions, building a foundation for more advanced mathematical concepts you'll encounter in the future. We'll make sure to clarify each step, so even if you're new to algebra, you'll feel confident following along and even tackling similar problems on your own.

Step 1: Eliminating the Fraction

The first hurdle in our equation, $8 = \frac{3}{2}(7x - 2)$, is the fraction $\frac{3}{2}$. To make things simpler, we want to get rid of this fraction. The most straightforward way to do this is by multiplying both sides of the equation by the reciprocal of $\frac{3}{2}$, which is $\frac{2}{3}$. This is a direct application of the multiplication property of equality, which states that if you multiply both sides of an equation by the same non-zero number, the equation remains true. By multiplying $\frac{3}{2}$ by its reciprocal $\frac{2}{3}$, we effectively get 1 on that side of the equation, leaving the expression $(7x - 2)$ unhindered. So, let's perform this operation: multiply 8 by $\frac{2}{3}$ and multiply $\frac{3}{2}(7x - 2)$ by $\frac{2}{3}$. On the left side, $8 \times \frac{2}{3} = \frac{16}{3}$. On the right side, $\frac{2}{3} \times \frac{3}{2}(7x - 2) = 1 \times (7x - 2) = (7x - 2)$. Now, our equation looks much cleaner: $\frac{16}{3} = 7x - 2$. This initial step is crucial because it removes a potential source of error and simplifies the subsequent calculations, allowing us to focus on isolating 'x' more directly. It's like clearing the path before a race; once the obstacles are removed, the rest of the journey becomes much smoother. Remember that in algebra, fractions can often be the most intimidating part of an equation, but understanding how to eliminate them using reciprocals is a fundamental skill that opens up many possibilities for solving complex problems.

Step 2: Isolating the Term with 'x'

We've successfully simplified our equation to $\frac{16}{3} = 7x - 2$. Now, our next objective is to get the term containing 'x' (which is $7x$) by itself on one side. Currently, it's stuck with a '- 2' next to it. To undo the subtraction of 2, we need to perform the inverse operation, which is addition. We will add 2 to both sides of the equation. This is based on the addition property of equality, which allows us to add the same quantity to both sides without changing the equation's balance. So, on the left side, we have $\frac{16}{3} + 2$. To add these, we need a common denominator. Since 2 can be written as $\frac{6}{3}$, we have $\frac{16}{3} + \frac{6}{3} = \frac{16 + 6}{3} = \frac{22}{3}$. On the right side, we have $-2 + 2$, which equals 0. So, the right side simply becomes $7x$. Our equation has now transformed into $\frac{22}{3} = 7x$. This step is vital because it separates the term with our variable from any constant terms, bringing us closer to isolating 'x'. It demonstrates how we use inverse operations systematically to peel away the numbers surrounding our variable. Each step in solving an algebraic equation is like peeling layers off an onion, gradually revealing the core. By adding 2 to both sides, we've successfully eliminated the constant term from the side with 'x', setting the stage for the final step in our solution.

Step 3: Solving for 'x'

We are in the home stretch! Our equation is now $\frac{22}{3} = 7x$. The only thing separating 'x' from being completely isolated is the coefficient 7, which is multiplying 'x'. To undo multiplication, we use its inverse operation: division. We need to divide both sides of the equation by 7. This again utilizes the division property of equality, ensuring the equation's balance is maintained. Dividing the right side, $7x$, by 7 gives us simply $x$. Now, for the left side, we have $\frac{22}{3}$ divided by 7. Dividing a fraction by a whole number is the same as multiplying the fraction by the reciprocal of that whole number. The reciprocal of 7 is $\frac{1}{7}$. So, we calculate $\frac{22}{3} \times \frac{1}{7}$. To multiply fractions, we multiply the numerators together and the denominators together: $22 \times 1 = 22$ and $3 \times 7 = 21$. Therefore, $\frac{22}{3} \div 7 = \frac{22}{21}$. So, our final solution is $x = \frac{22}{21}$. This final step completes the process of isolating 'x', giving us its exact value. It's a satisfying moment when you reach the end of a multi-step problem like this, knowing that each operation was performed correctly to arrive at the definitive answer. This method of using inverse operations—multiplication/division and addition/subtraction—is the bedrock of solving linear equations.

Step 4: Verification (Optional but Recommended)

To be absolutely sure our solution is correct, we can perform a verification step. This involves plugging our found value of $x = \frac{22}{21}$ back into the original equation $8 = \frac{3}{2}(7x - 2)$ and checking if both sides are equal. Let's substitute $x = \frac{22}{21}$ into the right side of the equation:

$7x - 2 = 7\left(\frac{22}{21}\right) - 2$

First, simplify $7\left(\frac{22}{21}\right)$. We can cancel out a factor of 7 from the 7 in the numerator and the 21 in the denominator ($21 = 7 \times 3$):

$7\left(\frac{22}{21}\right) = \frac{7}{1} \times \frac{22}{21} = \frac{1}{1} \times \frac{22}{3} = \frac{22}{3}$

Now, substitute this back into the expression:

$\frac{22}{3} - 2$

To subtract 2, we need a common denominator. We write 2 as $\frac{6}{3}$:

$\frac{22}{3} - \frac{6}{3} = \frac{22 - 6}{3} = \frac{16}{3}$

So, the expression inside the parentheses simplifies to $\frac{16}{3}$. Now, let's multiply this by the $\frac{3}{2}$ that was outside the parentheses in the original equation:

$\frac{3}{2} \times \frac{16}{3}$

We can cancel out the 3s, and we can simplify 16 and 2 by dividing both by 2:

$\frac{\cancel{3}}{2} \times \frac{16}{\cancel{3}} = \frac{1}{2} \times \frac{16}{1} = \frac{1 \times 16}{2 \times 1} = \frac{16}{2} = 8$

The right side of the equation evaluates to 8. Since the original equation stated that $8 = \frac{3}{2}(7x - 2)$, and we found that the right side equals 8 when $x = \frac{22}{21}$, our solution is indeed correct! This verification process is a critical habit to develop in mathematics, as it catches errors and builds confidence in your answers. It confirms that the value of 'x' we found makes the original statement a true mathematical fact.

Conclusion

We've successfully navigated the process of solving the equation $8 = \frac{3}{2}(7x - 2)$ for the variable $x$. Through a series of logical steps, involving the elimination of fractions, isolation of the variable term, and finally solving for $x$, we arrived at the solution $x = \frac{22}{21}$. This problem highlights the fundamental principles of algebraic manipulation, demonstrating how to use inverse operations and properties of equality to simplify and solve equations. Remember that practice is key to mastering these skills. The more equations you solve, the more comfortable you'll become with the process, and the faster you'll be able to identify the most efficient path to a solution. Don't be discouraged by complex-looking equations; they can often be broken down into simpler, manageable steps. Keep practicing, stay curious, and enjoy the journey of mathematical discovery!

For further exploration into algebraic equations and problem-solving techniques, you can visit resources like Khan Academy. Their comprehensive lessons and practice exercises are excellent for reinforcing your understanding of these concepts and developing your mathematical prowess.