8.8. Cramer’s Rule http://www.ck12.org
Example A
Represent the following system of equations as a matrix equation.
y− 13 =− 3 x
x= 19 − 4 ySolution: First write each equation in standard form.
3 x+y= 13
x+ 4 y= 19Then write as a coefficient matrix times a variable matrix equal to a solution matrix.[
3 1
1 4
]
·
[x
y]
=
[ 13
19
]
Example B
Solve the system from Example A using Cramer’s Rule.
Solution:
x=∣∣
∣∣ef bd∣∣
∣∣
∣∣
∣∣a b
c d∣∣
∣∣
=
∣∣
∣∣13 119 4
∣∣
∣∣
∣∣
∣∣3 1
1 4
∣∣
∣∣
=^133 ··^44 −−^191 · 1 ·^1 =^3311 = 3
y=∣∣
∣∣3 13
1 19
∣∣
∣∣
∣∣
∣∣3 1
1 4
∣∣
∣∣
=^3 ·^1911 −^13 =^4411 = 4
Example C
What isyequal to in the following system?
x+ 2 y−z= 0
7 x− 0 y+z= 14
0 x+y+z= 10Solution: If you attempted to solve this using elimination, it would take over a page of writing and rewriting to
solve. Cramer’s Rule speeds up the solving process.
1 2 − 1
7 0 1
0 1 1
·
x
y
z