256 Two-by-Two Linear Systems
Adding the two morphed equations in their entirety causes the variable t to vanish:
− 4 t+u=− 3
4 t+ 10 u=− 14
11 u=− 17
When we divide this result through by 11, we see that u=−17/11.
Eliminate the second variable
Now let’s look at the first original equation and the second morphed equation again.
They are
2 t+ 5 u=− 7
and
− 4 t+u=− 3
Our goal this time is to find a way to make u vanish when we add multiples of the equations.
Let’s multiply the second equation through by −5. That morphs it into
20 t− 5 u= 15
We can add the two equations in their entirety:
2 t+ 5 u=− 7
20 t− 5 u= 15
22 t= 8
This is easily solved to get t= 8/22, which reduces to 4/11. We can now state the solution to
this system as an ordered pair (t,u)= (4/11,−17/11).
Are you confused?
Let’s plug these results into the original equations to be sure they’re accurate. The first equation figures
out this way:
2 t+ 5 u=− 7
2 × 4/11 + 5 × (−17/11)=− 7
8/11 + (−85/11)=− 7
−77/11=− 7
− 7 =− 7