Pre-Calculus

Consider the chemical reaction aC2H6 + bCO2 + cH2O → dC2H5OH, where a, b, c, and d are unknown positive integers.?

The reaction must be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 2b+c=d. While there are many possible choices for a, b, c, and d that balance the reaction, it is customary to use the smallest possible integers. Balance this reaction.

if it helps, I'm learning about matrices, so if there is an easy way for me to do this on a calculator, I would really appreciate the help. =)

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  1. if not, then can you show me how to do this with the substitution method or something?

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  2. C: 2a+b=2d
    H: 6a+2c=6d
    O: 2b+c=d

    now,
    6a+3b=6d
    6a+2c=6d
    so, 3b=2c

    6a+3b=6d
    12b+9b=6d
    so, a=3b=2c
    So, if we choose a=6 because 6=LCM(2,3)

    a=6
    b=2
    c=3
    d=7

    and we have

    6C2H6 + 2CO2 + 3H2O → 7C2H5OH

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  3. I was not able to follow the steps at all.

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