To solve the equation, we start from:
\[ \frac{x}{5} - 12 = \frac{3}{y} \]
First, add 12 to both sides to isolate the fraction:
\[ \frac{x}{5} = \frac{3}{y} + 12 \]
Now, let's express 12 as a fraction with a denominator of \(y\):
\[ 12 = \frac{12y}{y} \]
Thus, we can rewrite the right side:
\[ \frac{x}{5} = \frac{3 + 12y}{y} \]
Next, we cross-multiply to eliminate the fractions:
\[ x \cdot y = 5(3 + 12y) \]
Distributing the 5:
\[ xy = 15 + 60y \]
Now we can rearrange this equation to isolate \(x\):
\[ xy - 60y = 15 \]
Factoring out \(y\):
\[ y(x - 60) = 15 \]
From here, we can find specific values for \(x\) and \(y\). Let's say:
-
Let \(y = 1\):
- \(1(x - 60) = 15\) → \(x - 60 = 15\) → \(x = 75\) So, one pair is \( (75, 1) \).
-
Let \(y = 3\):
- \(3(x - 60) = 15\) → \(x - 60 = 5\) → \(x = 65\) So, another pair is \( (65, 3) \).
-
Let \(y = 5\):
- \(5(x - 60) = 15\) → \(x - 60 = 3\) → \(x = 63\). So, another pair is \( (63, 5) \).
Based on the pairs provided, we could replace the missing fractions in our original equation like this:
Thus, we can conclude:
\[ 75/5 - 12 = 3/1 \]
or
\[ 65/5 - 12 = 3/3 \]
or
\[ 63/5 - 12 = 3/5 \]
All satisfy the equation, and thus the missing fractions could be represented as:
Final Answer:
75/5 - 12 = 3/1 (which is a valid option).
Any of the pairs will work.
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