Algebra 2/Trig
4.4
Solving Systems With Matrix Equations
p. 244
Learning Targets: Use matrices to solve systems of linear equations in mathematical and
real-world situations.
EXAMPLE 1:
Real Numbers:
4x = 16
How would you solve? We cannot divide
Matrices so we multiply by the inverse.
Solve 4x = 16 by multiplying by the inverse.
Write the matrix equation that represents each system.
2𝑥 + 𝑦 = 2
``{
5𝑥 − 3𝑦 = −17
AX = B We want to solve for x, we cannot divide, so we multiply by the inverse of A.
AX = B
A-1(AX) = BA-1
(A-1A)X = BA-1
IX = A-1B
X = A-1B
Must be in this order!
In the calculator: Matrix A, Matrix B
Find A-1B = X
X=
Therefore, x = _______________ and y = _________________.
1
EXAMPLE 2:
Solve, if possible.
{
2𝑦 − 10 = −8𝑥
5𝑥 + 𝑦 = 7
First, write in Standard Form:
Next, write the system of Matrices:
Use your calculator to solve for x and y.
EXAMPLE 3: Write the matrix equation and solve, if possible.
𝑥 + 𝑦 − 𝑧 = 14
{4𝑥 − 𝑦 + 5𝑧 = −22
2𝑥 + 2𝑦 − 3𝑧 = 35
Write the system of matrices:
To solve find: X = A-1B
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EXAMPLE 4: Write the matrix equation and solve.
3𝑥 − 2𝑦 + 5𝑧 − 𝑤 = −10
6𝑦 + 2𝑧 + 3𝑤 = 12
{
The order is x, y, z, w.
8𝑥 − 4𝑧 − 𝑤 = 0
−8𝑥 + 9𝑤 = −16
Write the system of matrices and solve. (Answers in fractions please.)
EXAMPLE 5:
A financial manger wants to invest $100,000 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 be invested at each interest rate to earn $10,000 in interest per year?
Let x represent the amount invested at 5% and let y represent the amount invested at 14%..
1.) Write a system of linear equations to represent the situation.
2.) Now write the system as a matrix equation and solve for x and y.
HW: 4.4 pages 248 – 250 Problems 12 – 30 EVEN and 32 – 36 ALL
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