Algebra Know-It-ALL

2. The second and third revised equations we obtained in the section “Eliminate One
   Variable” were

2 x− 5 y−z=− 1

and

3 x+ 6 y− 7 z= 0

The two-variable equation in x and y that we derived from these, as a result of eliminating

the variable z, was

− 11 x+ 41 y= 7

If we take this equation together with the solution to Prob. 1, we have the two-by-two

linear system

− 37 x− 4 y= 35

and

− 11 x+ 41 y= 7

3. First, let’s get rid of x. We can multiply the top equation through by −11 and the
   bottom equation through by 37. When we do these maneuvers and then add the
   resulting equations, we get

407 x+ 44 y=− 385

− 407 x+ 1,517y= 259
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
1,561y=− 126

Dividing through by 1,561 tells us that y=−126 / 1,561, which reduces to −18/223.

Now, let’s get rid of y. We can multiply the top equation through by 41 and the bottom

equation through by 4. When we do that and then add the results, we get

−1,517x− 164 y= 1,435

− 44 x+ 164 y= 28
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
−1,561x= 1,463

We divide through by −1,561 to get x= 1,463 / (−1,561), which reduces to −209/223.

The solution to this two-by-two linear system is therefore

x=−209/223

Chapter 18 651