Solve the equation: [tex]$\frac{5-6 y}{3}+\frac{y}{8}=0$[/tex]
Answer (2)
Multiply both sides of the equation by the least common multiple (LCM) of 3 and 8, which is 24, to eliminate the fractions.
Distribute and simplify the equation: 8 ( 5 − 6 y ) + 3 y = 0 ⇒ 40 − 48 y + 3 y = 0 .
Combine like terms to isolate the y term: 40 − 45 y = 0 .
Solve for y: y = 45 40 = 9 8 .
The final answer is 9 8 .
Explanation
Problem Analysis We are given the equation: 3 5 − 6 y + 8 y = 0 Our goal is to solve for y .
Eliminating Fractions To eliminate the fractions, we multiply both sides of the equation by the least common multiple (LCM) of 3 and 8. The LCM of 3 and 8 is 24. Multiplying both sides by 24, we get: 24 × ( 3 5 − 6 y + 8 y ) = 24 × 0 24 × 3 5 − 6 y + 24 × 8 y = 0 8 ( 5 − 6 y ) + 3 y = 0
Distributing Now, we distribute the 8 across the terms in the parenthesis: 8 × 5 − 8 × 6 y + 3 y = 0 40 − 48 y + 3 y = 0
Combining Like Terms Next, we combine like terms: 40 − 45 y = 0
Solving for y Now, we isolate the y term by subtracting 40 from both sides: − 45 y = − 40 Finally, we solve for y by dividing both sides by -45: y = − 45 − 40 y = 45 40 y = 9 8
Final Answer Therefore, the solution to the equation is: y = 9 8
Examples
Imagine you're baking a cake and need to adjust a recipe that uses fractions of ingredients. Solving equations like this helps you determine the exact amount of each ingredient needed to maintain the recipe's balance. For instance, if you want to scale a recipe up or down, you'll need to solve similar equations to find the new quantities of each ingredient. This ensures your cake turns out perfectly every time!
The solution to the equation 3 5 − 6 y + 8 y = 0 is y = 9 8 . This was found by eliminating fractions, simplifying, and isolating the variable. Overall, we used algebraic methods to manipulate the equation step by step.
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