4.4 Solving Systems With Matrix Equations

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Presentation transcript:

4.4 Solving Systems With Matrix Equations Objective: Use matrices to solve systems of linear equations in mathematical and real-world situations. Standard: 2.8.11.1. Use matrices to organize and manipulate data.

Solving a matrix equation of the form AX = B, where X = x y , is similar to solving a linear equation in the form ax = b, where a, b, and x are real numbers and a ≠ 0. Real Numbers Matrices ax = b AX = B ½ (ax) = ½(b) A-1(AX) = A-1B (1/a• a)x = b/a (A-1A)X = A-1B x = b/a IX = A-1B X = A-1B Just as 1/a must exist in order to solve ax = b (where a ≠ o), A-1 must exist to solve AX = B. CALCULATOR: A-1* B

Ex 1. 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. A). How much money should be invested at each interest rate to earn $5000 in interest per year? X + Y = 50,000 .05X + .14Y = 5,000

B). How much money should the manager invest at each interest rate to earn $4000 in interest per year? X + Y = 50,000 .05X + .14Y = 5,000 5%  $33,333.33 14%  $16,666.67

-4x + 2y – z = -3 2x – y + 3z = -6 -2x + 3y – 2z = 1 Ex 3. Refer to the system of equations at right. 2y – z = 4x - 3 a. Write the system as a matrix equation. 2x + 3z = y – 6 b. Solve the matrix equation. 3y – 1 = 2x + 2z -4x + 2y – z = -3 2x – y + 3z = -6 -2x + 3y – 2z = 1 X = 1 Y = -1 and Z = -3

-3x + 4y = 3 * Ex 4. Solve: -6x + 8y = 18 , if possible, by using a matrix equation. If not possible, classify the system.

9x - 3y = 27 * Ex 5. Solve: - 6x + 2y = -18 , if possible, by using a matrix equation. If not possible, classify the system.

Writing Activities 12). The system at the right can be represented by a matrix equation. What will be the dimensions of the coefficient matrix? Explain.