Let the system of equations

x + 5y - z = 1

4x + 3y - 3z = 7

24x + y + λz = μ

λ, μ ∈ ℝ, have infinitely many solutions. Then the number of the solutions of this system,

if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is :

<p>$$\begin{aligned} &\text { For infinitely many solution }\\ &\begin{aligned} & \Delta=0 \\ & \left|\begin{array}{ccc} 1 & 5 & -1 \\ 4 & 3 & -3 \\ 24 & 1 & \lambda \end{array}\right|=0 \\ & \Rightarrow 1(3 \lambda+3)-5(4 \lambda+72)-1(4-72)=0 \\ & \Rightarrow-17 \lambda+3-4 \times 72-4=0 \\ & \Rightarrow 17 \lambda=-289 \\ & \Rightarrow \lambda=-17 \\ & \Delta 1=0 \end{aligned} \end{aligned}$$</p> <p>$$\begin{aligned} & \Rightarrow\left|\begin{array}{ccc} 1 & 5 & -1 \\ 7 & 3 & -3 \\ \mu & 1 & -17 \end{array}\right|=0 \\ & \Rightarrow 1(-51+3)-5(-119+3 \mu)-1(7-3 \mu)=0 \\ & \Rightarrow-48+595-15 \mu-7+3 \mu=0 \\ & \Rightarrow 12 \mu=540 \\ & \mu=45 \\ & x+5 y-z=1 \\ & 4 x+3 y-3 z=7 \\ & 24 x+y-17 z=45 \\ & \text { Let } z=1 \\ & x+5 y=1+\lambda] \times 4 \\ & 4 x+3 y=7+3 \lambda \end{aligned}$$</p> <p>$\frac{\underset{-}{4 \mathrm{x}}+20 \mathrm{y}=\underset{-}{4+4 \lambda}}{-17 \mathrm{y}=3-\lambda}$</p> <p>$$\begin{aligned} \begin{aligned} \mathrm{y} & =\frac{\lambda-3}{17}, \mathrm{x}=1+\lambda-\frac{5 \lambda-15}{17} \\ & =\frac{32-12 \lambda}{17} \\ 7 & \leq \frac{\lambda-3}{17}+\frac{32+12 \lambda}{17}+\lambda \leq 77 \\ 7 & \leq \frac{30 \lambda+29}{17} \leq 77 \\ 3 & \leq \lambda \leq 42 \\ \lambda & =3,20,37 \end{aligned} \end{aligned}$$</p>