Accel Precalc Unit 3: Matrices Lesson 8: Solving Systems of Equations with Matrices EQ: How do you use matrices in the graphing calculator to solve systems of equations?

How do you enter matrices in your graphing calculator?

A system of equations may be represented as a matrix equation. For example, the system of equations may be represented by the matrix equation

Ex. 1 Write the matrix equation that represents the system:

Ex. 2 Write the matrix equation that represents the system:

Ex. 3 Write the system of equations represented by the matrix equation

A matrix equation is in the form AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Real Numbers ax=b (1/a)(ax) = (1/a)b (1/a)(a)x = b/a 1x = b/a x = b/a Solving AX=B Note: 1/a must exist to solve ax = b

Real Numbers ax=b (1/a)(ax) = (1/a)b (1/a)(a)x = b/a 1x = b/a x = b/a Matrices AX=B A -1 (AX)=A -1 B (A -1 A)X=A -1 B IX=A -1 B X=A -1 B Solving AX=B Note: A -1 must exist to solve AX=B

How do you use matrices to solve systems of equations?

Ex. 3 Solve the system of equations using matrices. x = -7 y = 15

Ex. 4 Solve the system of equations using matrices. x = 1 y = -1

Ex. 5 Solve the system of equations using matrices. x = -2 y = 4 z = 5

In Class Practice:

Ex. 5 Application A financial manager wants to invest $50,000 for a client by putting some of the money in a low-risk investment that earns 5% per year and some of the money in a high-risk investment that earns 14% per year. How much money should she invest at each interest rate to earn $5000 in interest per year? Let x = amount invested at 5% Let y = amount invested at 14% x + y = x +.14y = 5000

Write as a matrix equation to solve.

YES!!! Multiplication of matrices DOES NOT follow the commutative property.

x + y = x +.14y = 5000 Write as a matrix equation to solve. She should invest $22, at 5% and $27, at 14% to earn $5,000 per year.

Solve the system using a matrix equation. Matrix A does not have an inverse. (What is the determinant of A?) Determine if the system is inconsistent or dependent. This system is inconsistent (parallel lines) because the slopes are the same but the y-intercepts are different.

Ex. 7 Solve the system using a matrix equation. Matrix A does not have an inverse. (What is the determinant of A?) Determine if the system is inconsistent or dependent. This system is dependent (same line) because the slopes are the same and the y-intercepts are the same.

Assignment: Textbook p. 589 #51 – 63 odd Use the determinant and/or a matrix equation to solve each system. If there is not a unique solution, classify the system as consistent dependent or inconsistent.