For a system of linear equations to have a unique solution, certain conditions have to be met. This question discusses those conditions.

1. Linear Equation

2. Linear Equation
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.
Which of the following statements if true would imply that the above
system of equations does not have a unique solution?
i.
a1
a2
=
b1
b2
=
c1
c2
not equal to
d1
d2
ii.
a1
a2
=
a2
a3
;
b1
b2
=
b2
b3
iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3
are irrational numbers

3. Linear Equation
If we have three independent equations, we will have a unique
solution. In other words, we will not have unique solutions if
i. The equations are inconsistent or
ii. Two equations can be combined to give the third
Now, let us move to the statements.
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.
Which of the following statements if true would imply that the above
system of equations does not have a unique solution?

4. Linear Equation
Statement (i):
a1
a2
=
b1
b2
=
c1
c2
not equal to
d1
d2
This tells us that the first two equations cannot hold good at the
same time. Think about this x + y + z = 3; 2x + 2y + 2z = 5. Either the
first or the second can hold good. Both cannot hold good at the same
time. So, this will definitely not have any solution.
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.
Which of the following statements if true would imply that the above
system of equations does not have a unique solution?

5. Linear Equation
Statement (ii): a1, a2, a3 are in GP, b1, b2, b3 are in GP. This does not
prevent the system from having a unique solution.
For instance, if we have
x + 9y + 5z = 11
2x + 3y – 6z = 17
4x + y – 3z = 15
This could very well have a unique solution.
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.
Which of the following statements if true would imply that the above
system of equations does not have a unique solution?

6. Linear Equation
Statement (iii): a1, a2, a3 are integers; b1, b2, b3 are rational numbers,
c1, c2, c3 are irrational numbers. This gives us practically nothing. This
system of equations can definitely have a unique solution.
So, only Statement I tells us that a unique solution is impossible.
Answer choice (a)
a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.
Which of the following statements if true would imply that the above
system of equations does not have a unique solution?

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