Question

Consider the equation 3 E ≡ 5x − 2y = .
Write down equations E , respectively so that 1 E,
2 E,
3
i) E and E are inconsistent; 1
ii) E and E have a unique solution; 2
iii) E and E have infinitely many solutions.

Answer 1

For a system of linear equations in two variables, exactly one of the following is true.

$$\begin{matrix} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{matrix}$$

1. The system has no solution.

$$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=0\ but \ \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}=\not 0\ or \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}=\not0$$

$$\begin{matrix} E : 5x-2y=3 \\ E1: -10x+4y=5 \end{matrix}$$

2. The system has exactly one solution.

$$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=\not0$$

$$\begin{matrix} E : 5x-2y=3 \\ E2: x+4y=5 \end{matrix}$$

3. The system has infinitely many solutions.

$$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=0 , \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}=0, \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}=0$$

$$\begin{matrix} E : 5x-2y=3 \\ E3: -15x+6y=-9 \end{matrix}$$