Solving Systems of 3 or More Variables

14.6

Why a Matrix?

• In previous math classes you solved systems of two linear equations using the following method:• Graphing

• Substitution

• Elimination

• These solutions were 2-dimensional (x, y). Matrices can also be used to solve systems of equations. They are especially useful in systems that involve 3 or more variables.

We Live in a 3 Dimensional World, … Right?

• Add the following points to the graph• (3, 4, 0)and (3, 4, -2)

Look carefully at the example of graphing the 3 dimensional point (3,2,4).

Where x=3, y=2, and z=4

(3 ,4 ,0)(3 ,4 ,−2)

Solve by hand { −𝑥+𝑦 +2𝑧=32 𝑥− 𝑦+𝑧=3

−5 𝑥+2 𝑦+3 𝑧=4

Using a Matrix to solve

• First: Split the system into 3 matrices:

• Matrix A holds the coefficients

• Matrix X holds the variables• This matrix is called the vector matrix.

• Matrix B hold the integer values to the right of the = sign.

Ex 2:

Sove for

• To solve for “X”, we need to isolate matrix X by simplifying matrix A to “1” in matrix form.• We do this by multiplying Matrix A

by it inverse matrix

• So…to solve a system of equations using matrices we use:

Now Solve example 2 again, this time using a matrix and your calculator. [ 𝐴 ]−1 [ 𝐴 ] [ 𝑋 ]= [𝐴 ]− 1 [𝐵 ]

Now you try…

Aren’t you glad you aren’t doing this by hand?