The variables $x_1,x_2,x_3,x_4$ represent the number of molecules that combine to produce

the end result.

$($a$)$ Write a system of linear equations by writing an equation for each element $C,H,O$

so that the number of atoms on the left is equal to the number of atoms on the

right. For example, for $H$ to be balanced, $x_1=x_4.$

$($b$)$ Convert this system to a homogeneous augmented matrix.

$($c$)$ Reduce the augmented matrix to row echelon form and in the final matrix, circle the

pivot positions. All transformations must be written down. Example: $R_{1}harr{R}_2.$

$($d$)$ Write out the solution. Because this is a chemical equation, the solution must

have the smallest possible positive integers.