Q.55 For a = ________, the following simultaneous equations have an infinite number of solutions:
10𝑥𝑥 + 13𝑦𝑦 = 6
𝑎𝑎𝑥𝑥 + 32.5𝑦𝑦 = 15
The system of simultaneous equations 10x + 13y = 6 and ax + 32.5y = 15 has infinite solutions when a = 25, as this makes the equations proportional and coincident lines.[web:1][code_file:1]
Condition for Infinite Solutions
A pair of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 has infinite solutions if a₁/a₂ = b₁/b₂ = c₁/c₂.[web:3][web:8]
Here, rewrite as 10x + 13y - 6 = 0 and ax + 32.5y - 15 = 0, so coefficients are a₁=10, b₁=13, c₁=-6 and a₂=a, b₂=32.5, c₂=-15.[web:1]
This condition means the lines overlap completely, representing the same equation scaled by a constant.[web:2]
Detailed Calculation
- Compute ratios:
13/32.5 = 0.4and-6/-15 = 0.4.[code_file:1] - Set
10/a = 0.4, soa = 10/0.4 = 25.[code_file:1][web:12] - Verify:
10/25 = 0.4 = 13/32.5 = 6/15, confirming proportionality.[web:6]
Verification by Substitution
With a=25, second equation is 25x + 32.5y = 15.[web:6]
Multiply first by 2.5: 25x + 32.5y = 15, identical to second, yielding infinite solutions along the line.[web:4]
The determinant of the coefficient matrix is zero, and augmented matrix rank equals coefficient rank, supporting infinite solutions.[web:9]
Other Possibilities Explained
| Condition | Ratio Relationship | Result |
|---|---|---|
| Unique Solution | a₁/a₂ ≠ b₁/b₂ |
Exactly one solution[web:1] |
| No Solution | a₁/a₂ = b₁/b₂ ≠ c₁/c₂ |
Parallel lines[web:7] |
| Infinite Solutions | a₁/a₂ = b₁/b₂ = c₁/c₂ |
a = 25[web:5] |
