Multiple choice

The system of linear equations 4x + 2y = 7 and 2x + y = 6 has

  1. a unique solution

  2. no solution

  3. an infinite number of solutions

  4. exactly two distinct solutions
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\text{The given system is} \\ \begin{bmatrix} \ 4 & 2 \ \ 2 & 1 \ \end{bmatrix} \begin{bmatrix} \ x \ \ y \ \end{bmatrix} = \begin{bmatrix} \ 7 \ \ 6 \ \end{bmatrix} \\ \text{We have} \hspace{1cm} A = \begin{bmatrix} \ 4 & 2 \ \ 2 & 1 \ \end{bmatrix} \\ \text{and} \hspace{1cm} |A| = \begin{vmatrix} \ 4 & 2 \ \ 2 & 1 \ \end{vmatrix} = 0 \hspace{1cm} \text{Rank of matrix $\rho(A) < 2$} \\ \text{Now} \hspace{1cm} C = \begin{vmatrix} \ 4 & 2 & | & 7\ \ 2 & 1 & | & 6\ \end{vmatrix} \hspace{1cm} \text{Rank of matrix $\rho(C) = 2$} \\ \text{Since $\rho{A} \neq \rho (C)$ there is no solution.}$

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