Q.No. 46 The system of linear equations:
cx + y = 5
3x + 3y = 6
has no solution when c is equal to ____________________.
The system of linear equations cx + y = 5 and 3x + 3y = 6 has no solution when c = 1.
Condition for No Solution
A system of two linear equations a1x + b1y = c1 and
a2x + b2y = c2 has no solution if the lines are parallel, meaning:
a1/a2 = b1/b2 ≠ c1/c2
Here, equation 1 is cx + y = 5 so:
a1 = c, b1 = 1, c1 = 5
Equation 2 is 3x + 3y = 6 so:
a2 = 3, b2 = 3, c2 = 6
Simplify second equation by dividing by 3: x + y = 2.
Now system is cx + y = 5 and x + y = 2.
Ratios:
c/1 = 1/1 ≠ 5/2 ⇒ c = 1
Since 5/2 = 2.5 ≠ 1, no solution exists at c = 1.
Detailed Verification
If c = 1, equations become:
x + y = 5
3x + 3y = 6 → x + y = 2
Subtract: 0 = 3, contradiction → No solution.
If c ≠ 1 (example c = 2), solve:
From second, y = 2 − x
Plug into first: 2x + (2 − x) = 5 → x = 3, y = −1 → solution exists.
So only c = 1 makes coefficients proportional but constants not.
The system of linear equations no solution c value is a common IIT JAM math query. For cx + y = 5 and 3x + 3y = 6, no solution occurs specifically when c = 1.
No Solution Condition Explained
Parallel lines give no intersection when:
a1/a2 = b1/b2 ≠ c1/c2
Simplifying: c = 1 but 5/2 ≠ 1 ⇒ inconsistent.
Step-by-Step Solution for IIT JAM
Ratios: c/3 = 1/3 → c = 1.
Check constants: 5/6 ≠ 1/3 → inconsistent.
Verification
c = 1 yields x + y = 5 and x + y = 2 → impossible.
Exam Tips: Unique vs Infinite Solutions
| Condition | Ratio Equality | Example |
|---|---|---|
| No Solution | a1/a2 = b1/b2 ≠ c1/c2 | c = 1 here |
| Unique Solution | Ratios not equal | c = 2 gives unique |
| Infinite Solutions | a1/a2 = b1/b2 = c1/c2 | Not applicable here |
Master system of linear equations no solution for exams—practice ratio tests!
