To determine which system of equations corresponds to the given matrix:
[tex]\[
\left[\begin{array}{ll|l}
2 & -3 & 15 \\
5 & -8 & 31
\end{array}\right]
\][/tex]
we need to carefully examine the matrix and match it with the provided options for the systems of equations.
We can rewrite the represented format of the matrix into equations:
From the first row: [tex]\( [2, -3, 15] \)[/tex]
[tex]\[
2x - 3y = 15
\][/tex]
From the second row: [tex]\( [5, -8, 31] \)[/tex]
[tex]\[
5x - 8y = 31
\][/tex]
Now, let's compare these equations with the given options:
Option A:
[tex]\[
2x + 3y = 15
\][/tex]
[tex]\[
5x - 8y = 31
\][/tex]
This system doesn't match the matrix because [tex]\(2x + 3y = 15\)[/tex] is different from [tex]\(2x - 3y = 15\)[/tex].
Option B:
[tex]\[
x - 3y = 15
\][/tex]
[tex]\[
x - 8y = 31
\][/tex]
This system doesn't match the matrix either because [tex]\(x - 3y = 15\)[/tex] and [tex]\(x - 8y = 31\)[/tex] do not correspond to [tex]\(2x - 3y = 15\)[/tex] or [tex]\(5x - 8y = 31\)[/tex].
Option C:
[tex]\[
2x - 3y = 15
\][/tex]
[tex]\[
5x - 8y = 37
\][/tex]
The first equation in Option C matches the first equation derived from the matrix. However, the second equation, [tex]\(5x - 8y = 37\)[/tex], does not match the second equation derived from the matrix, [tex]\(5x - 8y = 31\)[/tex].
Since none of the given options directly correspond to both equations in the matrix, the correct determination is that none of the given sets of equations accurately represent the matrix. Thus, the answer is:
None of the options match the provided matrix.
