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 solutions of this system, if x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77, is:
Question: Let the system of equations { x + 5 y − z = 1 4 x + 3 y − 3 z = 7 24 x + y + λ z = μ , λ , μ ∈ R \begin{cases} x + 5y - z = 1 \\[1em] 4x + 3y - 3z = 7 \\[1em] 24x + y + \lambda z = \mu,\quad \lambda, \mu \in \mathbb{R} \end{cases} have infinitely many solutions. Then, the number of solutions of this system, if x , y , z x, y, z are integers and satisfy 7 ≤ x + y + z ≤ 77 , is: Question Image Short Solution Idea: For infinitely many solutions, the third equation must be a linear combination of the first two. Find λ \lambda and μ \mu . Reduce to two equations in x , y , z x, y, z , find the relation among variables. Put x + y + z = t x + y + z = t , derive bounds 7 ≤ t ≤ 77 Count integer solutions. Image Solution