In chemistry we can use a system of linear equations to determine the coefficients necessary to balance a chemical equation. For example, the complete combustion of octane (C8H18) with oxygen (O2) results in carbon dioxide (CO2) and water (H2O). Because the quantity of each element is conserved, we must have the same quantities of carbon (C)), oxygen (O) and hydrogen (H) on each side of the following chemical equation: a(C8H18) + b(O2) ? c(CO2) + d(H2O) where the coefficients a, b, c, d are how many of each molecule is present. (a) Write the equation describing the conserved quantity of each type of chemical. You will have 3 equations in 4 unknowns. The equation for conserved oxygen atoms is given as an example: 2b = 2c + d ? 2b - 2c - d = 0. (b) Write the augmented matrix for this system. (c) Use Gaussian elimination to find the solutions to this homogeneous system of three equations in four unknowns. Pick the free parameter in the solution so that a, b, c, d are each positive integers having no common factors. This will give you the usual balancing equation for complete combustion of octane. (d) The chemical equation above is for the complete combustion of octane and oxygen. In reality, the chemical reaction involves two steps and if there is not enough oxygen relative to octane, combustion will be incomplete. This results in the accumulation of the intermediate product carbon monoxide CO which will be created but not completely converted into carbon dioxide CO2. This is potentially dangerous in home heating. The incomplete combustion of octane and oxygen is described by: a(C8H18) + b(O2) ? c(CO2) + d(H2O) + e(CO). If we set a = 2 this equation becomes 2(C8H18) + b(O2) ? c(CO2) + d(H2O) + e(CO). Solve for c, d, e in terms of the parameter b. (e) Note that all four unknowns must be non-negative, so that b has a specific range of possible values. Describe that range and discuss how the varying b within the range affects the output of this chemical reaction.