Question

The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

For a fraction $a/b$ to be an integer, $b$ must be a factor of $a$.

Answer 1

The Correct Option is C

Step 1: Substitute $y$ into the first equation: $3x + 4(mx + 1) = 9$.
Step 2: $3x + 4mx + 4 = 9 \implies x(3 + 4m) = 5$.
Step 3: $x = \frac{5}{3 + 4m}$.
Step 4: For $x$ to be an integer, $(3 + 4m)$ must be a divisor of 5. Divisors are $\pm 1, \pm 5$.
Step 5: $3 + 4m = 1 \implies 4m = -2 \implies m = -1/2$ (Not an integer). $3 + 4m = -1 \implies 4m = -4 \implies m = -1$ (Integer). $3 + 4m = 5 \implies 4m = 2 \implies m = 1/2$ (Not an integer). $3 + 4m = -5 \implies 4m = -8 \implies m = -2$ (Integer).
Step 6: Two integral values of $m$: $\{-1, -2\}$.