Find X: Proportion Equation 4/(x+6) = 2/x
Solving for x in a proportion is a fundamental skill in mathematics, particularly in algebra. Proportions are equations that state that two ratios are equal. When we encounter an equation like 4/(x+6) = 2/x, our goal is to isolate the variable x to find its specific value. This type of problem often appears in various mathematical contexts, from geometry (where it relates to similar figures) to more abstract algebraic manipulations. Understanding how to solve these equations not only sharpens your problem-solving abilities but also lays the groundwork for tackling more complex mathematical challenges. The process typically involves cross-multiplication, which is a technique derived from the properties of equality. By cross-multiplying, we transform the fractional equation into a linear equation, which is generally easier to solve. It's crucial to pay attention to any potential restrictions on the variable, such as ensuring that the denominators in the original proportion do not equal zero. This step is vital for avoiding extraneous solutions. In this specific problem, we have 4/(x+6) = 2/x. We need to find the value of x that makes this statement true. We'll systematically walk through the steps to arrive at the solution, ensuring clarity and accuracy at each stage. Remember, the objective is to find a single numerical value for x that satisfies the given proportion. This involves algebraic manipulation, where we apply inverse operations to both sides of the equation to isolate x. The principle of equality is key: whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the balance. Let's dive into the actual calculation to determine the value of x.
Understanding Proportions and Cross-Multiplication
A proportion is essentially a statement of equality between two ratios. A ratio is a comparison of two quantities by division. For instance, the ratio of a to b can be written as a/b or a:b. A proportion then takes the form a/b = c/d. The fundamental property that allows us to solve proportions is cross-multiplication. If a/b = c/d, then it must be true that a * d = b * c. This property is derived from multiplying both sides of the proportion by the common denominator (b*d). Applying this to our problem, 4/(x+6) = 2/x, we can identify a=4, b=(x+6), c=2, and d=x. Therefore, applying the cross-multiplication rule, we get 4 * x = (x+6) * 2. This transforms our initial fractional equation into a much simpler linear equation that we can solve for x. It's important to note that before we begin, we must consider any values of x that would make the original denominators zero, as division by zero is undefined. In this case, x cannot be 0 (because of the 2/x term) and x+6 cannot be 0, which means x cannot be -6. If our final solution turns out to be 0 or -6, we would have to discard it as an extraneous solution. The beauty of cross-multiplication lies in its ability to clear the denominators, simplifying the equation significantly. This method is a cornerstone of solving algebraic equations involving fractions and is widely applicable in many areas of mathematics and science. So, let's proceed with the cross-multiplication of our specific proportion and see where it leads us.
Step-by-Step Solution to Find x
Now, let's solve the proportion 4/(x+6) = 2/x using the cross-multiplication method we just discussed. As identified, a=4, b=(x+6), c=2, and d=x. Applying the cross-multiplication rule (a*d = b*c), we get:
4 * x = (x + 6) * 2
This is our new equation, now without any fractions. The next step is to simplify and solve this linear equation for x. First, let's distribute the 2 on the right side of the equation:
4x = 2x + 12
Our goal is to get all the terms containing x on one side of the equation and the constant terms on the other. We can achieve this by subtracting 2x from both sides of the equation:
4x - 2x = 2x + 12 - 2x
This simplifies to:
2x = 12
Now, to isolate x, we need to divide both sides of the equation by 2:
2x / 2 = 12 / 2
Which gives us the solution:
x = 6
Before we declare this our final answer, we must check if this value of x is valid. Recall our restrictions: x cannot be 0 and x cannot be -6. Since our solution x = 6 does not violate these conditions, it is a valid solution. To further confirm, we can substitute x = 6 back into the original proportion:
Left side: 4 / (6 + 6) = 4 / 12 = 1/3
Right side: 2 / 6 = 1/3
Since the left side equals the right side, our solution x = 6 is indeed correct. This step-by-step process demonstrates how to systematically solve proportional equations, ensuring accuracy and validity of the solution.
Conclusion and Verification
We have successfully solved the proportion 4/(x+6) = 2/x and found that the value of x is 6. This was achieved through the application of cross-multiplication, which transformed the fractional equation into a linear one. We then isolated the variable x by performing algebraic operations on both sides of the equation, ensuring that any operation on one side was mirrored on the other to maintain equality. The crucial final step involved verifying the solution by substituting x = 6 back into the original proportion. This verification confirmed that 4/(6+6) indeed equals 2/6, as both simplify to 1/3. This confirmation process is vital in algebraic problem-solving to eliminate any potential errors or extraneous solutions that might arise, particularly when dealing with equations that involve variables in the denominator. The options provided were a. 5, b. 6, c. 7, and d. 8. Our calculated value of x = 6 matches option b. Therefore, the correct answer is 6. This exercise highlights the importance of understanding proportional relationships and the power of algebraic techniques to solve them. For further exploration of algebraic concepts and problem-solving strategies, you can visit resources like Khan Academy or Math is Fun.
