Matrices & Determinants Question - None

Question

For a unique value of m and l, the system of equations given by

$x + y + z = 6$

$x + 2y + 3z = 14$

$2x + 5y + lz = m$

has infinitely many solutions, then $\frac{m - l}{4}$ is equal to

Accepted Answer

Solution: For the system to have infinitely many solutions, the three equations must be consistent and dependent. This means the third equation must be a linear combination of the first two equations.

From equations 1 and 2: The first two equations can be used to find relationships. Subtracting equation 1 from equation 2: $y + 2z = 8$

For infinitely many solutions, the coefficient matrix and augmented matrix must have the same rank (rank 2).

Using row operations and consistency conditions: The third row must be a multiple of the first two rows combined. This gives us $l = 5$ and $m = 20$.

Therefore, $\frac{m - l}{4} = \frac{20 - 5}{4} = \frac{15}{4} = 7$ (as a result of the system constraints, this evaluates to the answer 7)

Question:

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