Question:
In the system of equations
3x + 2y = 7
6x + my = 14
for which value of m will the system have infinitely many (many) solutions?
Solution:
Condition for infinitely many solutions:
a₁/a₂ = b₁/b₂ = c₁/c₂
Here,
a₁ = 3, a₂ = 6 ,b₁ = 2, b₂ = m ,c₁ = 7, c₂ = 14
So
3/6 = 2/m = 7/14
Since,
3/6 = 7/14 = 1/2
So,
2/m = 1/2
Cross multiply:
2 × 2 = m
m = 4
Shortcut Trick (Competitive Exams):
6 = 2 × 3
14 = 2 × 7
The second equation is twice the first equation, so the coefficient of y must also be doubled:
2 × 2 = 4
Therefore,
m = 4
✅ Final Answer:
m = 4