Question:
The number of rational values of \(m\) for which the \(y\)-coordinate of the point of intersection of the lines \(3x + 2y = 10\) and \(x = my + 2\) is an integer is
The number of rational values of \(m\) for which the \(y\)-coordinate of the point of intersection of the lines \(3x + 2y = 10\) and \(x = my + 2\) is an integer is
Show Hint
Find divisors of the constant term to determine possible values.
Updated On: Apr 23, 2026
- 2
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The Correct Option is C
Solution and Explanation
Step 1: Formula / Definition}
\[ 3(my+2) + 2y = 10 \Rightarrow (3m+2)y = 4 \]
Step 2: Calculation / Simplification}
\[ y = \frac{4}{3m+2} \]
For \(y\) to be integer, \(3m+2\) must divide 4
\(3m+2 \in \{\pm 1, \pm 2, \pm 4\}\)
\(3m \in \{-1, -3, 0, -4, 2, -6\}\)
\(m \in \left\{-\frac{1}{3}, -1, 0, -\frac{4}{3}, \frac{2}{3}, -2\right\}\)
Rational \(m\): all 6 values are rational.
Step 3: Final Answer
\[ 6 \]
\[ 3(my+2) + 2y = 10 \Rightarrow (3m+2)y = 4 \]
Step 2: Calculation / Simplification}
\[ y = \frac{4}{3m+2} \]
For \(y\) to be integer, \(3m+2\) must divide 4
\(3m+2 \in \{\pm 1, \pm 2, \pm 4\}\)
\(3m \in \{-1, -3, 0, -4, 2, -6\}\)
\(m \in \left\{-\frac{1}{3}, -1, 0, -\frac{4}{3}, \frac{2}{3}, -2\right\}\)
Rational \(m\): all 6 values are rational.
Step 3: Final Answer
\[ 6 \]
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