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    IBHardAlgebra10–12

    The Value of k That Breaks a Linear System

    O Valor de k que Quebra um Sistema Linear

    Consider the system 2x+3y=72x + 3y = 7 kx+6y=11kx + 6y = 11 Find the value of kk for which the system does not have a unique solution, and say whether the system then has infinitely many solutions or none.

    Uniqueness of a two-by-two system is decided by one number attached to the coefficient matrix. That number tells you when uniqueness fails, but not which of the two failures happens — for that, compare the two equations as a whole.

    Solution

    Step 1 of 6

    1. 1.The coefficient matrix is (23k6)\begin{pmatrix} 2 & 3 \\ k & 6 \end{pmatrix}, with determinant: