**Applications to Chemistry**

** Application 1 **It takes three different ingredients A, B, and
C, to produce a certain chemical substance. A, B, and C have to be dissolved in
water separately before they interact to form the chemical. Suppose that the
solution containing A at 1.5 g/cm

** Solution **Let x, y, z be the
corresponding volumes (in cubic centimeters) of the solutions containing A, B,
and C. Then 1.5x is the mass of A in the first case, 3.6y is the mass of B, and
5.3z is the mass of C. Added together, the three masses should give 25.07 g. So

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The same reasoning applies to the other two cases. This gives the linear system

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The augmented matrix of this system is

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Reducing the above matrix would give the solution

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** Application 2 **Another typical application of linear systems
to chemistry is

“*mass is neither
created nor destroyed in any chemical reaction. Therefore balancing of
equations requires the same number of atoms on both sides of a chemical reaction.
The mass of all the reactants (the substances going into a reaction)
must equal the mass of the products (the substances produced by the
reaction).”*

As an example consider the following chemical equation

C_{2}H_{ 6} +
O_{2} → CO_{2} +
H_{2}O.

Balancing this chemical reaction** **means finding values of x, y, z and t so that the number of atoms of each
element is the same on both sides of the equation:

xC_{2}H_{ 6} +
yO_{2} → zCO_{2} + tH_{2}O.

This gives the following linear system:

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The general solution of the above system is

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Since we are looking for whole values of the variables x, y z, and t, choose x=2 and get y=7, z= 4 and t=6. The balanced equation is then:

2C_{2}H_{ 6} +
7O_{2} → 4CO_{2} + 6H_{2}O.

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