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Question
Total number of solutions to the system of linear equations 2x + 3y + z = 1, 4X + 6y + 2z = 3, 6x + 9y + 3z = 2 is ___
Total number of solutions to the system of linear equations 2x + 3y + z = 1, 4X + 6y + 2z = 3, 6x + 9y + 3z = 2 is
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Solution
Once you express the given equations in the form of matrices the determinant D = 0 and D1 = D2 = D3 = 0. So according to Cramer's rule there exists either infinite number of solutions or no solutions at all. But how to find if its no solution case or infinite solution case. Here is when u can see the advantage of relating this to 3d geometry. If you visualize the 3 planes they are all parallel to each other I.e there is no point of intersection for them... therefore there exists no solution for these equations. Alternately if there exists one value which satisfies all 3 equations then 1 = 3/2 = 2/3 which is absurd.
Once you express the given equations in the form of matrices the determinant D = 0 and D1 = D2 = D3 = 0. So according to Cramer's rule there exists either infinite number of solutions or no solutions at all. But how to find if its no solution case or infinite solution case. Here is when u can see the advantage of relating this to 3d geometry. If you visualize the 3 planes they are all parallel to each other I.e there is no point of intersection for them... therefore there exists no solution for these equations. Alternately if there exists one value which satisfies all 3 equations then 1 = 3/2 = 2/3 which is absurd.
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