Write the chemical equation

$CO + H_20 = H_2 + CO_2$ as an equation in ordered triples $(H, O, C)$ where $H$, $O$, $C$ are the number of hidrogen, oxigen, , and carbon atoms, respectively, in each molecule.

• a) Write the chemical equation $pC_3H_4O_3 + qO_2 = rCO_2 + sH_2O$ as an equation in ordered triples with unknown coefficients p, g, r, and s.
• b) Find the smallest positive integer solution for p, g, r, and s.

The given chemical equation is:

\text{CO} + \text{H}_2\text{O} \rightarrow \text{H}_2 + \text{CO}_2

Representing each molecule as ordered triples ((H, O, C)), where (H), (O), and (C) are the number of hydrogen, oxygen, and carbon atoms respectively:

(0, 1, 1) + (2, 1, 0) = (2, 0, 0) + (0, 2, 1)

This verifies that the chemical equation balances in terms of atoms.

Now for the second chemical equation:

pC_3H_4O_3 + qO_2 \rightarrow rCO_2 + sH_2O

We can write the corresponding system of equations in terms of ordered triples ((H, O, C)):

p(4, 3, 3) + q(0, 2, 0) = r(0, 2, 1) + s(2, 1, 0)

Separating the equations for (H), (O), and (C):

\begin{align*}
\text{For H:} & \quad 4p = 2s \\
\text{For O:} & \quad 3p + 2q = 2r + s \\
\text{For C:} & \quad 3p = r
\end{align*}

To find the smallest positive integer solution for (p), (q), (r), and (s), solve the system of equations:

1. From $(\text{H}: 4p = 2s \quad \Rightarrow \quad s = 2p)$,
2. From $(\text{C}: 3p = r \quad \Rightarrow \quad r = 3p)$,
3. Substitute $(s = 2p)$ and $(r = 3p) into (\text{O}: 3p + 2q = 2(3p) + 2p)$:

   3p + 2q = 6p + 2p \quad \Rightarrow \quad 2q = 5p \quad \Rightarrow \quad q = \frac{5p}{2}.

To ensure all coefficients are integers, let (p = 2) (smallest positive integer):

p = 2, \quad q = 5, \quad r = 6, \quad s = 4.

Thus, the smallest positive integer solution is:

p = 2, \quad q = 5, \quad r = 6, \quad s = 4.

The balanced equation with the smallest positive integer coefficients is:

2C_3H_4O_3 + 5O_2 \rightarrow 6CO_2 + 4H_2O

---

Sistemas Generadores en Kⁿ

• Respuesta

Un Sistema compatible determinado puede tener más ecuaciones independientes que incógnitas. ¿Verdadero o Falso? Justifica tu respuesta.

Estudiar si el sistema de ecuaciones paramétricas

⎧	𝑥₁＝      𝜆 −  𝜇
⎨	𝑥₂＝ 2 + 3𝜆 +  𝜇
⎪	𝑥₃＝ 3 -  𝜆 + 2𝜇
⎩	𝑥₄＝     2𝜆 + 5𝜇

es equivalente (tiene las mismas soluciones) al sistema de ecuaciones implícitas

⎰  7𝑥₁ -  𝑥₂ + 4𝑥₃       ＝ 10 
⎱ 13𝑥₁ - 7𝑥₂       + 4𝑥₄ ＝ 10