Question

which set of simultaneous equations is generated by the matrix equation above?\\(\

$$\begin{bmatrix} 3 & -2 \\\\ 1 & 4 \\end{bmatrix}$$

\

$$\begin{bmatrix} x \\\\ y \\end{bmatrix}$$

= \

$$\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix}$$

\\)\
a. \\(\

$$\begin{cases} 3x - 2y = 1 \\\\ x + 4y = 2 \\end{cases}$$

\\)\
b. \\(\

$$\begin{cases} 3x + 2y = 1 \\\\ x - 4y = 2 \\end{cases}$$

\\)\
c. \\(\

$$\begin{cases} x + 4y = 2 \\\\ 3x - 2y = 2 \\end{cases}$$

\\)\
d. \\(\

$$\begin{cases} x + 2y = 1 \\\\ 4x + y = 2 \\end{cases}$$

\\)

Explanation:

Step1: Recall matrix multiplication

For a matrix equation \(

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

=

$$\begin{bmatrix} e \\ f \end{bmatrix}$$

\), the resulting system of equations is \(ax + by=e\) and \(cx + dy = f\).

Given the matrix equation \(

$$\begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$$

$$\begin{bmatrix} x \\ y \end{bmatrix}$$

=

$$\begin{bmatrix} 1 \\ 2 \end{bmatrix}$$

\), we apply matrix multiplication.

Step2: Multiply the matrices

First equation: \(3x+(- 2)y = 1\) which simplifies to \(3x - 2y=1\).

Second equation: \(1x + 4y=2\) which simplifies to \(x + 4y = 2\).

Now we check the options:

• Option a: \(3x - 2y = 1\) and \(x + 4y=2\) which matches our derived equations.
• Option b: First equation is \(3x + 2y=1\) (incorrect sign for \(y\) coefficient), so eliminate.
• Option c: First equation is \(3x-2y = 2\) (incorrect right - hand side), so eliminate.
• Option d: First equation is \(x + 2y=1\) (incorrect \(x\) coefficient) and second equation is \(4x + y=2\) (incorrect coefficients), so eliminate.

Answer:

a. \(3x - 2y = 1\)
\(x + 4y = 2\)