8.2. Systems of Three Equations and Three Unknowns http://www.ck12.org
Step 5: Subtract row 2 from row 3.
3 x+ 2 y+z= 7
0 x− 40 y− 16 z=− 72
0 x+ 0 y+ 11 z= + 22Step 6: Scale row 3 to find z.
3 x+ 2 y+z= 7
0 x− 40 y− 16 z=− 72
0 x+ 0 y+z= 2Now thatz=2, rewrite the system so it becomes a system of three equations and three unknowns. Any iteration of
the first two rows will work. This iteration is from step 6.
3 x+ 2 y+ 2 = 7
0 x− 40 y− 32 =− 72Concept Problem Revisited
When solving a system of three equations with three unknowns, you are allowed to add and subtract rows, swap
rows and scale rows. These three operations should allow you to eliminate the coefficients of the variables in a
systematic way.
Vocabulary
Swapping rowsmeans rewriting the system so that two rows change places. This is purely a visual reorganization
and should help you problem solve and see what to do next.
Eliminating the variablemeans making the coefficient equal to zero thereby removing that variable from that
particular equation and reducing the number of unknown quantities.
Scaling a rowmeans multiplying every coefficient in the row by any number you choose (besides zero). This can
be helpful for getting coefficients to match so that they can be eliminated.
Consistent systemsare systems that have at least one solution. If the equations arelinearly independent, then the
system will have just one solution.
Inconsistent systemsare systems that have no solution. An example isx= 2 ,x=3.
Guided Practice
- When Kaitlyn went to the store with ten dollars she saw that she had some choices about what to buy. She could
get one apple, one onion and one basket of blueberries for 9 dollars. She could get two apples and two onions for