1. EXAMPLE 3
Use Cramer’s rule for a 2 X 2 system
Use Cramer’s rule to solve this system:
3x 5y = 21
9x + 4y = 6 –
SOLUTION
STEP 1
Evaluate the determinant of the coefficient matrix. 9 4 3 –5
= –45 – 12 = –57

2. EXAMPLE 3
Use Cramer’s rule for a 2 X 2 system
STEP 2
Apply Cramer’s rule because the determinant is not 0.
x = –6 4
–21 –5
–57 =
–57
30 – (–84) =
–57
114 = –2
y = 9 –6 3
–21
–57 =
–57
–189 – (–18) =
–57
–171 3
ANSWER
The solution is ( 2, 3). –

3. Use Cramer’s rule for a 2 X 2 system
EXAMPLE 3
Use Cramer’s rule for a 2 X 2 system
CHECK
Check this solution in the original equations.
9x + 4y = –6
3x – 5y = –21
9(–2) + 4(3) = –6 ? ?
3(–2) – 5(3) = –21
– = –6 ?
–6 – 15 = –21 ? –6 =
–21 = –21

4. EXAMPLE 4
Solve a multi-step problem
CHEMISTRY
The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon (C), hydrogen (H), and oxygen (O).

5. EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Write a linear system using the formula for each compound. Let C, H, and O represent the atomic weights of carbon, hydrogen, and oxygen.
6C + 12H + 6O = 180
C O = 44
2H + 2O = 34

6. EXAMPLE 4
Solve a multi-step problem
STEP 2
Evaluate the determinant of the coefficient matrix. 6 1 12 2
= ( ) ( ) = 36 –
STEP 3
Apply Cramer’s rule because the determinant is not 0.
180 34 12 6 2 1 44
C =
H =
O = 36 –

7. EXAMPLE 4
Solve a multi-step problem
576 36 – =
432 = = =
ANSWER
The atomic weights of carbon, hydrogen, and oxygen are 12, 1, and 16, respectively.

8. GUIDED PRACTICE
for Examples 3 and 4
Use Cramer’s rule to solve the linear system.
5. 3x – 4y = –15
6. 4x + 7y = 2
–3x – 2y = 28
7. 3x – 4y + 2z = 18
2x + 5y = 13
4x + y – 5z = –13
2x – 3y + z = 11
(–1, 3).
ANSWER
(4, –2).
ANSWER
(2, –1, 4).
ANSWER