The system of equations 3x + 4y = 5, 6x + 7y = −8 is written in matrix form as
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}3 & 4 \\ 6 & 7\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}5 \\ -8\end{bmatrix}\)
A system of linear equations is written as AX = B, where each row of the coefficient matrix A holds the coefficients of one equation, X is the column of unknowns and B is the column of constants:
\( \displaystyle A = \begin{bmatrix}3 & 4 \\ 6 & 7\end{bmatrix},\quad X = \begin{bmatrix}x \\ y\end{bmatrix},\quad B = \begin{bmatrix}5 \\ -8\end{bmatrix} \)
Check by multiplying row by column:
\( \displaystyle \begin{bmatrix}3 & 4 \\ 6 & 7\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}3x + 4y \\ 6x + 7y\end{bmatrix} = \begin{bmatrix}5 \\ -8\end{bmatrix} \)
which gives back both equations.
Why the other options are incorrect
Option (1) \(\begin{bmatrix}3 & 4 \\ 6 & 7\end{bmatrix}\begin{bmatrix}x & y\end{bmatrix}=\begin{bmatrix}5 & -8\end{bmatrix}\) — Incorrect. A 2 × 2 matrix times a 1 × 2 matrix is not defined.
Option (2) \(\begin{bmatrix}3 & 6 \\ 4 & 7\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}=\begin{bmatrix}5 \\ -8\end{bmatrix}\) — Incorrect. The product is 3x + 6y and 4x + 7y, not the given equations.
Option (4) \(\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}3 & 4 \\ 6 & 7\end{bmatrix}=\begin{bmatrix}5 & -8\end{bmatrix}\) — Incorrect. The product is [3x + 6y 4x + 7y], not the given equations.