288 Larger Linear Systems
It agrees again! This practically guarantees that our answers are correct. We’ve found x and y by two differ-
ent routes, and they’ve come out the same both times. If we made an error somewhere, the answers would
almost certainly disagree.
Substitute Back
Now we have the values for x and y, and we’re confident that they’re correct because we’ve
arrived at them from two different directions. Here they are again:
x=−209/223
and
y=−18/223
Plug them in
We can use any of the original equations or their revisions to solve for z. Let’s use the third
original equation:
3 x=− 6 y+ 7 z
Plugging in the numbers for x and y, and proceeding step-by-step, we get
3 × (−209/223)=− 6 × (−18/223)+ 7 z
−627/223= 108/223 + 7 z
−735/223= 7 z
7 z=−735/223
z=−105/223
Are you confused?
If the last step in the above calculation confuses you, note that −735/7=−105. That’s the numerator in
the fraction. I had a feeling that −735 would cleanly divide by 7, because I acted on my hunch that z
would be a fraction with a denominator of 223. By now, you have probably noticed that in linear systems,
fractional solutions have a way of coming out with identical denominators. There’s a good reason for that,
but it would be a distraction to delve into the “why” of it right now. The important thing is that we have
all three solutions to our original three-by-three linear system, at least tentatively:
x=−209/223
y=−18/223
z=−105/223