Algebra Know-It-ALL

260 Two-by-Two Linear Systems

Are you confused?

As always, we had better check our work to be sure the solutions we obtained satisfy both of the original

equations. Here’s the first check:

− 7 v+w+ 10 = 0

− 7 × (2/3) + (−16/3)+ 10 = 0

−14/3− 16/3 + 10 = 0

−30/3+ 10 = 0

− 10 + 10 = 0

0 = 0

All right! Here’s the second check:

4 v+ 8 w=− 40

4 × 2/3 + 8 × (−16/3)=− 40

8/3 − 128/3 =− 40

(8 − 128)/3 =− 40

−120/3=− 40

− 40 =− 40

All right again! Our solutions are correct.

Here’s a challenge!

Solve the following pair of equations as a two-by-two linear system using the substitution method:

3 x−πy=− 1

and

− 8 x+ 2 y= 4

This process is going to be messy! If we remember that π is a plain old real number, it won’t be too bad.

The signs will be tricky, though.

Solution

Let’s tackle the second equation and get it into a form that expresses y in terms of x. First, we can add 8x

to each side, getting

2 y= 8 x+ 4

When we divide through by 2, we get

y= 4 x+ 2