The Row That Wasn't in the Matrix

An electronics workshop assembles two circuit boards, Board A and Board B. Last week's entire production is recorded as a single matrix equation:

$\begin{pmatrix} 4 & 3 \\ 2 & 5 \end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 27 \\ 31 \end{pmatrix}$

Here $a$ is the number of Board A units built and $b$ is the number of Board B units built. Each row of the matrix is one resource:

Row	Resource	Hours per Board A	Hours per Board B	Total hours used
$1$	Soldering station	$4$	$3$	$27$
$2$	Testing rig	$2$	$5$	$31$

The workshop now wants to cost out a third resource that was never written into the matrix. Packing takes $5$ hours per Board A and $2$ hours per Board B.

How many packing hours did last week's production require?

Answer in hours. The answer is a whole number — give it exactly.

The Row That Wasn't in the Matrix

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