Linear Equations - Conditions for Unique Solutions

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 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. = = not equal to ii. = ; = 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  

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 The equations are inconsistent or Two equations can be combined to give the third Now, let us move to the statements. 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?

4. Linear Equation Statement ( i ): = = not equal to 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.   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?

5. Linear Equation Statement (ii): a 1 , a 2 , a 3 are in GP, b 1 , b 2 , b 3 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. 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?

6. Linear Equation Statement (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. Answer choice (a) 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?

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