\frac{5}{x}-\frac{3}{2 x}=\frac{7}{6}
Answer (2)
Combine the fractions on the left side: x 5 − 2 x 3 = 2 x 7 .
Set up the equation: 2 x 7 = 6 7 .
Cross-multiply: 42 = 14 x .
Solve for x : x = 14 42 = 3 . The final answer is 3 .
Explanation
Problem Analysis We are given the equation x 5 − 2 x 3 = 6 7 Our goal is to solve for x .
Combining Fractions First, we combine the fractions on the left side of the equation. To do this, we need a common denominator, which is 2 x . So we rewrite the fractions: x 5 = x × 2 5 × 2 = 2 x 10 Now we can subtract the fractions: 2 x 10 − 2 x 3 = 2 x 10 − 3 = 2 x 7 So our equation becomes: 2 x 7 = 6 7
Cross-Multiplication Now, we can solve for x by cross-multiplying: 7 × 6 = 7 × 2 x 42 = 14 x
Solving for x Finally, we divide both sides by 14 to isolate x :
x = 14 42 = 3 So the solution is x = 3 .
Examples
Imagine you're baking a cake and need to adjust the ingredient quantities based on the number of servings. If the original recipe feeds a certain number of people and you want to scale it up or down, you'll use equations similar to this one to find the new amounts of each ingredient. Solving for x in such equations helps you determine the correct proportions to maintain the cake's taste and texture, ensuring it turns out perfectly whether you're baking for a small gathering or a large party. This type of problem is also applicable in chemistry when you need to calculate the amount of reactants needed for a chemical reaction.
To solve the equation x 5 − 2 x 3 = 6 7 , combine the fractions to get 2 x 7 = 6 7 and then cross-multiply. This leads to 42 = 14 x , giving the solution x = 3 .
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