Solving 1/5(k-3) = 3/4 A Step-by-Step Guide
Hey guys! Today, we're diving deep into the fascinating world of algebra to tackle a seemingly simple yet potentially tricky equation: 1/5(k-3) = 3/4. This equation falls under the realm of linear equations, and mastering it is a fundamental step towards conquering more complex mathematical challenges. So, buckle up and let's embark on this journey of mathematical discovery together!
Understanding the Equation: A Breakdown
Before we jump into the solution, let's take a moment to dissect the equation and understand its components. The equation 1/5(k-3) = 3/4 presents a scenario where a fraction, 1/5, is multiplied by an algebraic expression, (k-3). This entire product is then equated to another fraction, 3/4. Our mission, should we choose to accept it (and we do!), is to isolate the variable 'k' and determine its value.
At its core, this equation embodies the principle of equality. The left-hand side (LHS), 1/5(k-3), must be equivalent to the right-hand side (RHS), 3/4. Think of it as a balanced scale – if we perform an operation on one side, we must perform the same operation on the other side to maintain the equilibrium. This fundamental concept will guide us throughout the solution process.
The expression (k-3) within the parentheses represents a subtraction operation. The variable 'k' has 3 subtracted from it. The parentheses themselves indicate that this subtraction must be performed before we multiply by 1/5, adhering to the order of operations (PEMDAS/BODMAS). This is a crucial detail to keep in mind as we proceed.
On the other side of the equation, we have the fraction 3/4. This represents a proportion or a ratio. It signifies that for every 4 units, we have 3 units. Understanding this fractional representation is essential for manipulating the equation effectively.
In essence, the equation 1/5(k-3) = 3/4 presents a puzzle. We have an unknown value, 'k', hidden within the algebraic expression. By applying the principles of algebra and the order of operations, we can systematically unravel this puzzle and reveal the value of 'k'.
Step-by-Step Solution: Cracking the Code
Now that we have a solid understanding of the equation, let's dive into the step-by-step solution process. We'll break down each step with clarity and precision, ensuring that you grasp the underlying logic and reasoning.
Step 1: Eliminating the Fraction Outside the Parentheses
The first hurdle we encounter is the fraction 1/5 multiplying the expression (k-3). To simplify the equation, we need to get rid of this fraction. The most effective way to do this is to multiply both sides of the equation by the reciprocal of 1/5, which is 5. Remember, whatever we do to one side, we must do to the other to maintain the balance.
Multiplying both sides by 5, we get:
5 * [1/5(k-3)] = 5 * (3/4)
On the left-hand side, the 5 and 1/5 cancel each other out, leaving us with:
(k-3) = 5 * (3/4)
On the right-hand side, we multiply 5 by 3/4. This can be visualized as 5/1 * 3/4. Multiplying the numerators (5 * 3) and the denominators (1 * 4), we get 15/4.
So, our equation now looks like this:
(k-3) = 15/4
We've successfully eliminated the fraction outside the parentheses, making the equation more manageable. This step showcases the power of using reciprocals to simplify equations involving fractions.
Step 2: Isolating the Variable 'k'
Our next objective is to isolate the variable 'k' on one side of the equation. Currently, 'k' has 3 subtracted from it. To undo this subtraction, we need to perform the opposite operation: addition. We'll add 3 to both sides of the equation.
Adding 3 to both sides, we get:
(k-3) + 3 = 15/4 + 3
On the left-hand side, the -3 and +3 cancel each other out, leaving us with:
k = 15/4 + 3
Now, we need to add 3 to 15/4. To do this, we need to express 3 as a fraction with a denominator of 4. We can do this by multiplying 3 by 4/4, which is equivalent to 1.
3 = 3 * (4/4) = 12/4
Now we can add the fractions:
k = 15/4 + 12/4
Adding the numerators (15 + 12) while keeping the denominator the same, we get:
k = 27/4
We have successfully isolated the variable 'k'! The solution to the equation is k = 27/4.
Step 3: Verifying the Solution (Optional but Recommended)
To ensure that our solution is correct, it's always a good practice to verify it. We can do this by substituting k = 27/4 back into the original equation and checking if the left-hand side equals the right-hand side.
Original equation: 1/5(k-3) = 3/4
Substituting k = 27/4:
1/5(27/4 - 3) = 3/4
First, we need to subtract 3 from 27/4. Again, we'll express 3 as a fraction with a denominator of 4:
3 = 12/4
So, 27/4 - 3 = 27/4 - 12/4 = 15/4
Now, we substitute this back into the equation:
1/5(15/4) = 3/4
Multiplying 1/5 by 15/4, we get:
(1/5) * (15/4) = 15/20
Simplifying the fraction 15/20 by dividing both the numerator and denominator by 5, we get:
15/20 = 3/4
Therefore, the left-hand side (3/4) equals the right-hand side (3/4), confirming that our solution, k = 27/4, is indeed correct.
Alternative Approaches: Exploring Different Paths
While we've solved the equation using a specific method, it's worth noting that there are often alternative approaches to solving algebraic equations. Exploring these different paths can deepen your understanding and enhance your problem-solving skills. Let's briefly discuss one alternative approach for this equation.
Distributive Property First
Instead of multiplying both sides by 5 in the first step, we could have chosen to apply the distributive property to the left-hand side of the equation. The distributive property states that a(b + c) = ab + ac. In our case, this would mean multiplying 1/5 by both 'k' and '-3'.
Applying the distributive property to 1/5(k-3), we get:
(1/5) * k + (1/5) * (-3) = k/5 - 3/5
So, our equation would become:
k/5 - 3/5 = 3/4
From here, we would add 3/5 to both sides to isolate the term with 'k':
k/5 = 3/4 + 3/5
To add the fractions on the right-hand side, we need a common denominator, which is 20. Converting the fractions, we get:
k/5 = 15/20 + 12/20 = 27/20
Finally, we multiply both sides by 5 to isolate 'k':
k = (27/20) * 5 = 27/4
As you can see, this alternative approach, while slightly different, leads us to the same solution: k = 27/4. This demonstrates that there's often more than one way to skin a mathematical cat!
Common Pitfalls and How to Avoid Them
Solving algebraic equations can be a rewarding endeavor, but it's also easy to stumble along the way. Let's highlight some common pitfalls that students often encounter and discuss strategies for avoiding them.
Forgetting the Order of Operations
As mentioned earlier, the order of operations (PEMDAS/BODMAS) is paramount in algebra. Failing to adhere to this order can lead to incorrect results. In our equation, it's crucial to perform the subtraction within the parentheses (k-3) before multiplying by 1/5. Similarly, when simplifying expressions, remember to handle parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right) in that specific order.
Incorrectly Applying the Distributive Property
The distributive property is a powerful tool, but it must be applied correctly. A common mistake is forgetting to distribute to every term within the parentheses. For instance, in the alternative approach we discussed, we multiplied 1/5 by both 'k' and '-3'. If we had only multiplied by 'k', we would have arrived at an incorrect equation.
Errors with Fractions
Fractions can often be a source of confusion for students. When adding or subtracting fractions, remember that you need a common denominator. When multiplying fractions, multiply the numerators and the denominators. And when dividing fractions, remember to invert and multiply. Mistakes in these fractional operations can derail your solution.
Not Maintaining Balance in the Equation
The fundamental principle of equality dictates that whatever operation you perform on one side of the equation, you must perform on the other side. Failing to do so will disrupt the balance and lead to an incorrect solution. Always double-check that you're applying the same operation to both sides.
Skipping the Verification Step
While optional, verifying your solution is a crucial step in ensuring accuracy. Substituting your solution back into the original equation can reveal any errors you might have made along the way. It's a small investment of time that can save you from a lot of frustration.
Real-World Applications: Where Does This Math Show Up?
You might be wondering,
