For a system of equations to have infinite solutions, the determinant of the coefficient matrix must be zero.
The coefficient matrix is:
A=112−3−2−754λ
Expanding the determinant along Row 1:
det(A)=1[(−2)(λ)−(4)(−7)]−(−3)[(1)(λ)−(4)(2)]+5[(1)(−7)−(−2)(2)]
=1[−2λ+28]+3[λ−8]+5[−7+4]
=−2λ+28+3λ−24−15
=λ+28−24−15
=λ−11
Setting the determinant equal to zero:
λ−11=0
λ=11