# SYSTEMS of EQUATIONS (Gaussian Elimination)

```SYSTEMS of EQUATIONS
(Gaussian Elimination)
A system of linear equations is a set of one or more linear equations.
To solve a system of linear equations means to determine values that
will simultaneously satisfy the unknowns in each equation.
Ex 
Solve the following linear system using elimination:
2𝑥 + 𝑦 = −9
𝑥−𝑦 =6
Ex 
Determine the value(s) of a and b for which the given system has
no solution, an infinite number of solutions, and one solution:
𝑥 + 4𝑦 = 𝑎
𝑥 + 𝑏𝑦 = 8
A linear system can be written without its variables in matrix form.
A matrix is a rectangular array of numbers.
Ex 
Express the given system in matrix form:
2𝑥 + 𝑦 − 𝑧 = 3
5𝑥 + 𝑧 = 2
2𝑦 + 4𝑧 = −3

2 1 −1 3
5 0 1
2
0 2 4 −3
AUGMENTED MATRIX
The system can be solved using elementary row operations on the augmented matrix.
This procedure is called row reduction. It produces a series of equivalent systems.
ROW OPERATIONS:

Any row can be multiplied or divided by
a non–zero constant.

Any row can be replaced by the sum or difference
of that row and a multiple of another row.

Any two rows can be interchanged.
Ex 
Solve and classify each system:
a)
𝑥 + 2𝑦 + 𝑧 = −4
𝑥 + 4𝑦 + 5𝑧 = −18
4𝑥 − 𝑧 = −4
b)
𝑥 + 2𝑦 + 3𝑧 = −4
𝑥 − 𝑦 − 3𝑧 = 8
2𝑥 + 𝑦 + 6𝑧 = −14
HOMEWORK: p.507–509 #3, 5, 6, 9, 12(omit d), 14, 15
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