Algebra Know-It-ALL

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