Solve the Two-by-Two
Let’s continue our quest to solve the three-by-three linear system. We have obtained a two-by-
two system in x and y. Here it is again:
− 10 x+ 17 y= 8
and
− 11 x+ 41 y= 7
We can use any of the methods described in Chap. 16 to tackle this. Let’s use double elimina-
tion.
Eliminate x, solve for y
Let’s multiply the first equation through by −11 and the second equation through by 10.
When we do that, we get
110 x− 187 y=− 88
and
− 110 x+ 410 y= 70
Now let’s add these two equations in their entirety:
110 x− 187 y=− 88
− 110 x+ 410 y= 70
223 y=− 18
Dividing through by 223 tells us that y=−18/223. This fraction happens to be in lowest
terms (a fact that you can verify if you like).
Eliminate y, solve for x
Let’s multiply the first equation in our two-by-two system through by −41 and the second
equation through by 17. That gives us
410 x− 697 y=− 328
and
− 187 x+ 697 y= 119
Solve the Two-by-Two 285
