Question

Which system of linear equations can be solved using equation [x y] = [1/4,3/4,1,2] [28,-12]

Answer 1

The correct option is a.

The correct system of linear equations that can be solved using the given equation is:

`$-8x + 3y = 28$`

`$4x - y = -12$`

To determine which system of linear equations can be solved using the given equation
`$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right]$`, let's perform matrix multiplication to find the values of x and y:

`$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right]$`

Step 1: Multiply the matrices.

`$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{cc}(1)/(4) & (3)/(4) \\ 1 & 2\end{array}\right]\left[\begin{array}{c}28 \\ -12\end{array}\right] = \left[\begin{array}{c}(1)/(4) \cdot 28 + (3)/(4) \cdot (-12) \\ 1 \cdot 28 + 2 \cdot (-12)\end{array}\right]$`

Step 2: Calculate the values.

`$\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{c}7 - 9 \\ 28 - 24\end{array}\right] = \left[\begin{array}{c}-2 \\ 4\end{array}\right]$`

So, the values of x and y are
`$x = -2$` and
`$y = 4$`.

Now, let's check which system of equations corresponds to this solution.

System 1:

`$(1)/(4) x + y = 28$`

`$(3)/(4) x + 2y = -12$`

Plugging in the values we found:

`$(1)/(4) (-2) + 4 = -0.5 + 4 = 3.5 \\eq 28$` (Not true)

`$(3)/(4) (-2) + 2(4) = -1.5 + 8 = 6.5 \\eq -12$` (Not true)

System 2:

`$-8x + 3y = 28$`

`$4x - y = -12$`

Plugging in the values we found:

`$-8(-2) + 3(4) = 16 + 12 = 28$` (True)

`$4(-2) - 4 = -8 - 4 = -12$` (True)

Answer 2

The question is illustrated in matrix format.

To solve it we would need to proof that the value of x and y in the matrix corresponds with the answer