Inverse Matrices and Systems Section 4.7 Inverse Matrices and Systems

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

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

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

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

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

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

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

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

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.

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

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

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