Algebra Know-It-ALL

286 Larger Linear Systems

Adding these equations, we get

410 x− 697 y=− 328

− 187 x+ 697 y= 119

223 x=− 209

Dividing through by 223 tells us that x=−209/223. This fraction, like the previous one with

the same denominator, is in lowest terms.

Two down, one to go!

We’ve now solved for two of the three unknowns in our three-by-three system. We have the

values for x and y:

x=−209/223

and

y=−18/223

In the next section, we’ll substitute these values back into one of the original three equations

and solve for z. Then we’ll check our work. Something tells me that z is going to be a fraction

with a denominator of 223. What do you think?

Are you confused?

You might again question the choice of solution processes. “Why,” you might ask, “do we use the double-

elimination method to solve the two-by-two system here? Why not use morph-and-mix or rename-and-

replace?” The answer is, of course, “We can use either of those methods.”

Here’s a challenge!

Solve the preceding two-by-two system using the morph-and-mix method. Consider x the dependent

variable.

Solution

Once again, here’s our pair of equations:

− 10 x+ 17 y= 8

and

− 11 x+ 41 y= 7

We must get both of these into SI form, with x all by itself on the left sides of the equals signs. Step-by-step,

the first equation morphs like this:

− 10 x+ 17 y= 8

− 10 x=− 17 y+ 8