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 .
2 1
Explanation
Problem Analysis 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, we get: 5 ( 12 y − 1 ) = 2 ( 9 y + 8 )
Expanding the Equation 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. We subtract 18 y from both sides: 60 y − 18 y − 5 = 18 y − 18 y + 16 42 y − 5 = 16
Further Isolation 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
Conclusion Therefore, the value of y that makes the equation true is 2 1 .
Examples
In real-world scenarios, solving linear equations like this can help determine the break-even point in business. For example, if y represents the number of units sold, the equation could represent the point where the revenue from selling y units equals the cost of producing y units. Finding the value of y helps businesses understand how many units they need to sell to start making a profit. This concept is crucial for financial planning and decision-making in various industries.
The value of y that makes the equation true is 2 1 . This was found by eliminating fractions, expanding the equation, and isolating y . Therefore, the correct answer is option B: 2 1 .
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