The question is about checking for consistency of the given equations. Knowing a few equations, we can find whether the equations have an unique solution or infinite solutions or no solution. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 9: a_{1}x + b_{1}y + c_{1}z = d_{1}, a_{2}x + b_{2}y + c_{2}z = d_{2}, a_{3}x + b_{3}y + c_{3}z = d_{3}.

Which of the following statements if true would imply that the above system of equations does not have a unique solution?

i. \\frac{a_{1}}{a_{2}}\\) = \\frac{b_{1}}{b_{2}}\\) = \\frac{c_{1}}{c_{2}}\\) ≠ \\frac{d_{1}}{d_{2}}\\)

ii. \\frac{ a_{1} }{ a_{2} }\\)= \\frac{ a_{2} }{ a_{3} }\\) ; \\frac{ b_{1} }{ b_{2} }\\)= \\frac{ b_{2} }{ b_{3} }\\)

iii. a_{1}, a_{2}, a_{3} are integers; b_{1}, b_{2}, b_{3} are rational numbers, c_{1}, c_{2}, c_{3} are irrational numbers

- Statement i
- Statement ii
- Statement iii
- None

If we have three independent equations, we will have a unique solution. In other words, we will not have unique solutions if:

The equations are inconsistent or

Two equations can be combined to give the third

Now, let us move to the statements.

i.\\frac{ a_{1} }{ a_{2} }\\) = \\frac{ b_{1} }{ a_{2} }\\) = \\frac{ c_{1} }{ c_{2} }\\) ≠ \\frac{ d_{1} }{ d_{2} }\\)

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.

ii.\\frac{ a_{1} }{ a_{2} }\\)= \\frac{ a_{2} }{ a_{3} }\\) and \\frac{ b_{1} }{ b_{2} }\\) = \\frac{ b_{2} }{ b_{3} }\\)

a_{1}, a_{2}, a_{3} are in GP, b_{1}, b_{2}, b_{3}. 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.

iii. a_{1}, a_{2}, a_{3} are integers; b_{1}, b_{2}, b_{3} are rational numbers, c_{1}, c_{2}, c_{3} 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.

The question is **"Which of the following statements if true would imply that the above system of equations does not have a unique solution?"**

Statement i is the correct answer.

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