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(1) The country of Guilder has the following industries: oil-drilling, electricity-generation, and
truck manufacturing. The production of $1 of oil requires $0.1 of oil, $0.30 of electricity and
$0.1 worth of trucks for its transportation. The production of $1 of electricity requires $0.25
of oil, $0.4 of electricity, and $0.15 of trucks. Finally, the production of $1 worth of trucks
requires $0.2 of oil, $0.5 of electricity, and $0.1 of trucks. Assume also that during a period
of one month, Guilder wants to export $50,000 worth of oil, $75,000 worth of electricity, and
$125,000 worth of trucks.
Find the production level of each of these industries in that period of one month required
to exactly meet both the production and the export demands..
(2) Determine if the columns of
$\begin{bmatrix} 5 & -7 & -4 & 9 \\ 6 & -8 & -7 & 5 \\ 4 & -4 & -9 & -9 \\ -9 & 11 & 16 & 7 \end{bmatrix}$
span $\mathbb{R}^4$.
(3) Give the solutions to the following system in parametric vector form:
$x_1 + 3x_2 + x_3 = 1$
$-4x_1 - 9x_2 + 2x_3 = -1$
$-3x_2 - 6x_3 = -3$
