Algebra Know-It-ALL

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