Solve.
[tex]\frac{2}{x}=\frac{6}{2-x}[/tex]
Answer (2)
Cross-multiply to eliminate fractions: 2 ( 2 − x ) = 6 x .
Expand and simplify the equation: 4 − 2 x = 6 x ⇒ 4 = 8 x .
Solve for x : x = 8 4 = 2 1 .
Verify the solution by substituting x = 2 1 back into the original equation, which confirms the solution. The final answer is 2 1 .
Explanation
Problem Analysis We are given the equation x 2 = 2 − x 6 and we want to solve for x . This equation involves fractions, so our first goal is to eliminate the fractions by cross-multiplying.
Cross-Multiplication Cross-multiplying means multiplying both sides of the equation by x and by ( 2 − x ) . This gives us: 2 ( 2 − x ) = 6 x
Expanding the Equation Next, we expand the left side of the equation by distributing the 2: 4 − 2 x = 6 x
Isolating x Now, we want to isolate x on one side of the equation. We can add 2 x to both sides: 4 = 6 x + 2 x 4 = 8 x
Solving for x To solve for x , we divide both sides by 8: x = 8 4
Simplifying the Solution Finally, we simplify the fraction: x = 2 1
Checking the Solution To make sure our solution is correct, we can plug x = 2 1 back into the original equation: 2 1 2 = 2 − 2 1 6 4 = 2 3 6 4 = 6 ⋅ 3 2 4 = 4 Since the equation holds true, our solution is correct.
Final Answer Therefore, the solution to the equation x 2 = 2 − x 6 is x = 2 1 .
Examples
Imagine you're adjusting a recipe that serves 6 people, but you only want to make enough for 2. If one ingredient calls for x 2 cups, and the adjusted recipe needs 2 − x 6 cups, solving this equation helps you determine the exact amount of that ingredient you need. This kind of proportional adjustment is common in cooking, scaling designs, or managing resources where maintaining ratios is crucial.
The solution to the equation x 2 = 2 − x 6 is found to be x = 2 1 by cross-multiplying and isolating x . This solution has been verified by substituting back into the original equation. Therefore, x = 2 1 is indeed correct.
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