1. Inverse Matrices and Systems
Section 4.7
Inverse Matrices and Systems

2. Solving Systems of Equations Using Inverse Matrices
System of Equations:
𝟓𝒙+𝟐𝒚=𝟓 𝟑𝒙+ 𝒚=𝟏𝟒
Matrix Equation:
𝟓 𝟐 𝟑 𝟏 𝒙 𝒚 = 𝟓 𝟏𝟒
x-coefficients
y-coefficients
variables
constants

3. Example 1
Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix.
3𝑥+2𝑦=16 𝑦=5

4. Example 2
Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix.
−𝑏+2𝑐=4 𝑎+𝑏−𝑐=0 2𝑎+3𝑐=11

5. Example 3
Solve each system.
2𝑥+3𝑦=11 𝑥+2𝑦=6

6. Example 4
Solve each system.
5𝑎+3𝑏=7 3𝑎+2𝑏=5

7. Example 5
Solve each system.
2𝑥+𝑦+3𝑧=1 5𝑥+𝑦−2𝑧=8 𝑥−𝑦−9𝑧=5

8. TOTD
5𝑥+𝑦=14 4𝑥+3𝑦=20
Use matrices and write out each step.

9. Example 6
Solve each system.
𝑥+𝑦+𝑧=2 2𝑥+𝑦=5 𝑥+3𝑦−3𝑧=14

10. Example 7
Business A bead store has a sale on certain beads. Find the price of each size bead. Three small beads and two large beads are $3.25. Four small beads and three large beads cost $4.75.

11. Example 8
Write a coefficient matrix for each system. Use it to determine whether the system has a unique solution.
𝑥+𝑦=3 𝑥−𝑦=7

12. Example 9
Write a coefficient matrix for each system. Use it to determine whether the system has a unique solution.
𝑥+2𝑦=5 2𝑥+4𝑦=8

13. TOTD
Find the determinant of the coefficient matrix to determine whether the system has a unique solution.
If there is a unique solution, use a matrix equation to find that solution.
𝑥−3𝑦=−1 −6𝑥+19𝑦=6