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

A system of two linear equations in two variables has the form

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

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

1. The system has no solution.

$$\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=0,\ but \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}=\not0,\ or\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}=\not0$$

$$\begin{matrix} E: \ 5x-2y=3 \\ E1: -15x+6y=-5 \end{matrix}$$

2. The system has exactly one solution.

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

$$\begin{matrix} E: 5x-2y=3 \\ E2: x+6y=-5 \end{matrix}$$

3. The system has infinitely many solutions.

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

$$\begin{matrix} E: 5x-2y=3 \\ E3: 10x-4y=6 \end{matrix}$$