Question

Given that ([-1,2],[2,1])([x],[y])=([3],[4]), what are the values of x and y ?

a. x=1,y=2
b. x=-2,y=-1
c. x=2,y=1
d. x=-1,y=-2

Answer 1

Upon carefully reviewing the system of linear equations given by the matrix multiplication, the values of x and y were determined to be x = 1 and y = 2, corresponding to option a. A previous miscalculation was identified and corrected during this comprehensive review of the solution.

Step-by-step explanation:

The student has presented a matrix equation that needs to be solved for the variables x and y. The equation can be represented as:

\[\begin{bmatrix}-1 & 2\\2 & 1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}3\\4\end{bmatrix}\]

To find x and y, we need to solve this system of linear equations:

• \(-1x + 2y = 3\)
• \(2x + y = 4\)

Solving these linear equations, we obtain:

• Multiply the second equation by 2: \(4x + 2y = 8\)
• Add this to the first equation: \(-1x + 4x + 2y + 2y = 3 + 8\)
• Simplify to get \(3x + 4y = 11\)
• Now, let's solve the previous equations for x and y.

First equation: \(-1(x) + 2(y) = 3\) becomes \(-1x = 3 - 2y\)

Substitute \(y = 4 - 2x\) into the above equation to get \(-1x = 3 - 2(4 - 2x)\), which simplifies to \(-1x = 3 - 8 + 4x\), leading to \(5x = -5\) and thus \(x = -1\).

Substitute \(x = -1\) back into the second equation \(2x + y = 4\), we get \(2(-1) + y = 4\), which simplifies to \(y = 4 + 2 = 6\), which is not consistent with the given choices. This indicates there might have been a miscalculation; let's correct it.

Correcting the error, by substituting \(x = -1\) into \(2x + y = 4\), we actually get \(2(-1) + y = 4\) which simplifies to \(-2 + y = 4\), so \(y = 4 + 2 = 6\).

However, upon reviewing this, it appears that our initial system of equations was not accurately solved. We must undertake the process again to find the correct solution.

Let's revisit the system:

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

Solving the first equation for \(x\), we get \(x = 2y - 3\).

Substitute \(x = 2y - 3\) into the second equation:

\(2(2y - 3) + y = 4\)

\(4y - 6 + y = 4\)

\(5y = 10\)

\(y = 2\)

Now, substitute \(y = 2\) back into the first equation:

\(-1x + 2(2) = 3\)

\(-1x + 4 = 3\)

\(-1x = -1\)

\(x = 1\)

The values of x and y are therefore x = 1 and y = 2, which corresponds to option a