APPLIED MATHEMATICS 1A (MATH132) ADDITIONAL PROBLEMS 1 (2025) 1. Suppose that the augmented matrix for a system of linear equations has been reduced to the following matrix \[ \left[\begin{array}{ccccc|c} 1 & 5 & -4 & 0 & -7 & -5 \\ 0 & 0 & 1 & 1 & 7 & 3 \\ 0 & 0 & 0 & 1 & 4 & 2 \end{array}\right] \] Find the general solution of the linear system. (Ans: \( x_{1}=-1-5 x_{2}-5 x_{5}, x_{2} \) is free, \( x_{3}=1-3 x_{5}, x_{4}=2-4 x_{5}, x_{5} \) is free) 2. Consider the system \[ \begin{aligned} x_{1}+2 x_{3} & =a \\ 4 x_{1}-x_{2}+6 x_{3} & =b \\ -5 x_{1}+3 x_{2}-4 x_{3} & =c . \end{aligned} \] What condition must \( a, b \) and \( c \) satisfy for the system to be consistent? (Ans: \( c= \) \( 7 a-3 b) \)

The process of reducing an augmented matrix into reduced row echelon form is foundational in linear algebra. For the given matrix, we can use back-substitution to express leading variables in terms of free variables, ultimately leading us to the general solution. Every choice of values for the free variables \( x_2 \) and \( x_5 \) allows us to generate whole families of solutions, showcasing the infinite nature of systems with free variables! For the second problem, consistency of a system of equations can hinge upon the relationships between the coefficients of the variables and the constants on the right-hand side. When you transform the equations into a matrix form, you can derive conditions like \( c = 7a - 3b \) through operations that involve linear combinations of the rows. This shows how intricately interwoven the equations are, forming a delightful dance of dependencies!