Suppose that $k$ is a fixed, but unknown, positive integer. Suppose also that the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at the point with positive integer coordinates $(x, y)$. Since $9x+4y=600$ and $kx-4y=24$, adding these equations, we get $9x+kx=624$ and so $(9+k)x=624$. Since $x$ and $y$ are to be positive integers and $k>0$, then $9+k$ and $x$ are a positive divisor pair of 624 with $9+k>9$. Now $624=6 \cdot 104=6 \cdot 8 \cdot 13=2^{4} 3^{1} 13^{1}$, and so the positive divisors of 624 are $1,2,3,4,6,8,12,13,16,24,26,39,48,52,78,104,156,208,312,624$. We also want the value of $y$ to be a positive integer. Since the point $(x, y)$ lies on the line with equation $9x+4y=600$, then $4y=600-9x$ which gives $y=150-\frac{9}{4}x$, which is an integer exactly when $x$ is a multiple of 4. Therefore, we want $x$ to be a positive divisor of 624 which is a multiple of 4. Thus, the possible values of $x$ are $4,8,12,16,24,48,52,104,156,208,312,624$. The corresponding values of $9+k$ are $156,78,52,39,26,13,12,6,4,3,2,1$. Since $9+k>9$, we eliminate $6,4,3,2,1$ from this list. Thus, the possible values of $9+k$ are $156,78,52,39,26,13,12$. The corresponding values of $k$ are $147,69,43,30,17,4,3$. These correspond to the following values of $x: 4,8,12,16,24,48,52$. Using $y=150-\frac{9}{4}x$, these give the following values of $y: 141,132,123,114,96,42,33$. These are indeed all positive. This means that there are 7 values of $k$ with the required properties.