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Step-by-Step Solution
Step 1: Write down the system of equations
We are given the following system of linear equations:
1) $2x + 2y + 3z = a$ (P1)
2) $3x - y + 5z = b$ (P2)
3) $x - 3y + 2z = c$ (P3)
Step 2: Condition for the system to have more than one solution
A system of three variables can have infinitely many solutions if the three planes (represented by the equations) are not fully independent. In other words, one equation can be expressed as a linear combination of the other two.
Step 3: Check for linear dependence
Observe that if adding (P1) and (P3) yields (P2), then the three planes are not independent, which leads to infinitely many solutions. Let's add (P1) and (P3):
(P1) + (P3):
$(2x + 2y + 3z) + (x - 3y + 2z) = a + c.$
This simplifies to:
$3x - y + 5z = a + c.$
Compare this with (P2): $3x - y + 5z = b.$
Thus, for (P1) + (P3) to be equal to (P2), we must have:
$a + c = b.$
Step 4: Relate the constants a, b, and c
From $a + c = b$, we rearrange to get:
$b - c - a = 0.$
Therefore, the condition for the system to have more than one solution (i.e., infinitely many solutions) is
$b - c - a = 0.$
Conclusion
Hence, the correct relation among $a, b,$ and $c$ for the system of equations to have more than one solution is:
$b - c - a = 0.$
