Question

State Cramer’s rule. Show graphically how a system of two equations in 

two unknowns has:

(i) no solution, (ii) a unique, (iii) infinitely many solutions.

Answer 1

Cramer’s rule:

The solution of the simultaneous equations

$\begin{cases} a_1x+b_1y=d_1 \\ a_2x+b_2y=d_2 \end{cases}$

is given by the formulae

$x=\frac{\begin{vmatrix} d_1 & b_1 \\ d_2 & b_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$ $y=\frac{\begin{vmatrix} a_1 & d_1 \\ a_2 & d_2 \end{vmatrix} }{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$

(i) $\begin{cases} x-y=0 \\ x-y=1 \end{cases}$

This system has no solution.

ii)$\begin{cases} x-y=0 \\ x+y=0 \end{cases}$ 

This system has a unique solution.

(iii) $\begin{cases} x-y=1 \\ 2x-2y=2 \end{cases}$ 

This system has infinitely many solutions.