Q.55 For a = ________, the following simultaneous equations have an infinite number of solutions:
10π₯π₯ + 13π¦π¦ = 6
πππ₯π₯ + 32.5π¦π¦ = 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 aβx + bβy + cβ = 0 and aβx + bβy + cβ = 0 has infinite solutions if aβ/aβ = bβ/bβ = cβ/cβ.[web:3][web:8]
Here, rewrite as 10x + 13y - 6 = 0 and ax + 32.5y - 15 = 0, so coefficients are aβ=10, bβ=13, cβ=-6 and aβ=a, bβ=32.5, cβ=-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 | aβ/aβ β bβ/bβ |
Exactly one solution[web:1] |
| No Solution | aβ/aβ = bβ/bβ β cβ/cβ |
Parallel lines[web:7] |
| Infinite Solutions | aβ/aβ = bβ/bβ = cβ/cβ |
a = 25[web:5] |
Β
