Question #86900
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.
1
Expert's answer
2019-03-28T12:09:34-0400

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


a1x+b1y=c1a2x+b2y=c2\begin{matrix} a_1x+b_1y=c_1 \\ a_2x+b_2y=c_2 \end{matrix}

1. The system has no solution.


a1b1a2b2=0 but c1b1c2b2= ora1c1a2c2=\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

E:5x2y=3E1:10x+4y=5\begin{matrix} E : 5x-2y=3 \\ E1: -10x+4y=5 \end{matrix}

2. The system has exactly one solution.


a1b1a2b2=\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}=\not0

E:5x2y=3E2:x+4y=5\begin{matrix} E : 5x-2y=3 \\ E2: x+4y=5 \end{matrix}

3. The system has infinitely many solutions.


a1b1a2b2=0,c1b1c2b2=0,a1c1a2c2=0\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

E:5x2y=3E3:15x+6y=9\begin{matrix} E : 5x-2y=3 \\ E3: -15x+6y=-9 \end{matrix}


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