Select the correct answer.
Which value of [tex]$y$[/tex] makes this equation true?
[tex]\frac{12 y-1}{2}=\frac{9 y+8}{5}[/tex]
A. [tex]$\frac{1}{6}$[/tex]
B. [tex]$\frac{1}{2}$[/tex]
C. 2
D. 6
Answer (2)
Multiply both sides of the equation by 10 to eliminate the fractions: 5 ( 12 y − 1 ) = 2 ( 9 y + 8 ) .
Expand both sides of the equation: 60 y − 5 = 18 y + 16 .
Simplify and isolate y : 42 y = 21 .
Solve for y : y = 2 1 .
Explanation
Understanding the Problem We are given the equation 2 12 y − 1 = 5 9 y + 8 and asked to find the value of y that makes the equation true.
Eliminating Fractions To solve for y , we first eliminate the fractions by multiplying both sides of the equation by the least common multiple of 2 and 5, which is 10: 10 ⋅ 2 12 y − 1 = 10 ⋅ 5 9 y + 8
Simplifying the Equation Simplifying both sides, we get: 5 ( 12 y − 1 ) = 2 ( 9 y + 8 )
Expanding Both Sides Next, we expand both sides of the equation: 60 y − 5 = 18 y + 16
Isolating y Now, we want to isolate y on one side of the equation. Subtract 18 y from both sides: 60 y − 18 y − 5 = 18 y − 18 y + 16 42 y − 5 = 16
Further Isolating y Add 5 to both sides: 42 y − 5 + 5 = 16 + 5 42 y = 21
Solving for y Finally, divide both sides by 42: 42 42 y = 42 21 y = 2 1
Final Answer Therefore, the value of y that makes the equation true is 2 1 .
Examples
Imagine you're adjusting a recipe. The original recipe feeds a certain number of people, but you need to adjust the ingredients to feed a different number. This problem is similar to solving an equation where you need to find the right amount of an ingredient ( y ) to keep the recipe balanced. Understanding how to solve such equations ensures your adjustments are accurate, preventing your dish from being too salty or too bland. This skill is useful not only in cooking but also in various fields like finance and engineering, where proportional adjustments are frequently required.
The value of y that makes the equation 2 12 y − 1 = 5 9 y + 8 true is 2 1 . Therefore, the correct choice is option B. This was found by eliminating the fractions and isolating y step-by-step.
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