Answer:
D. The system of equations has infinite solutions.
Step-by-step explanation:
Given system of equations,
[tex]\begin{bmatrix}2 & 4 \\ 1 & 2\end{bmatrix}.\begin{bmatrix}x\\ y\end{bmatrix}=\begin{bmatrix}6\\ 3\end{bmatrix}[/tex]
[tex]\implies \begin{bmatrix}2x+4y\\ x+2y\end{bmatrix}=\begin{bmatrix}6\\ 3\end{bmatrix}[/tex]
By comparing both sides,
We get,
2x + 4y = 6,
x + 2y = 3,
We know that a system [tex]a_1x+b_1y=c_1[/tex], [tex]a_2x+b_2y=c_2[/tex]
has a unique solution if,
[tex]\frac{a_1}{a_2}\neq \frac{b_1}{b_2}\neq \frac{c_1}{c_2}[/tex]
No solution, if,
[tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}\neq \frac{c_1}{c_2}[/tex]
Infinitely many solution if,
[tex]\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}[/tex]
Here,
[tex]\frac{2}{1}=\frac{4}{2}=\frac{6}{3}=2[/tex]
Hence, the system of equation has infinite solutions.
Option D is correct.
