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How many values of $k$ allow the system of equations $(k+1)x + 8y = 4k$ and $kx + (k+3)y = 3k-1$ to have no solution?
For no solution, the determinant of the coefficient matrix must be zero but the system must be inconsistent.
$\Delta = (k+1)(k+3) - 8k = k^2+4k+3-8k = k^2-4k+3 = (k-1)(k-3) = 0$
So $k=1$ or $k=3$. Check each:
$k=1$: Equations become $2x+8y=4$ and $x+4y=2$ โ same line (infinite solutions). Not "no solution".
$k=3$: Equations become $4x+8y=12$ and $3x+6y=8$. Simplify: $x+2y=3$ and $x+2y=8/3$ โ contradiction! No solution.
So exactly $\mathbf{1}$ value of $k$ gives no solution.
The Threshold Limit Value (TLV), established by the American Conference of Governmental Industrial Hygienists (ACGIH), refers to the maximum concentration of a chemical or physical agent in the workplace air to which workers can be repeatedly exposed, day after day, without adverse health effects.
Types of TLV:
- TLV-TWA: Time-Weighted Average (8-hour workday, 40-hour week)
- TLV-STEL: Short-Term Exposure Limit (15-minute exposure)
- TLV-C: Ceiling value โ should never be exceeded
Understanding TLVs is essential for occupational health nurses managing industrial settings in Pakistan.
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