Understanding Linear Systems Equivalence with Gaussian Elimination and Row Operations

When are two linear systems called equivalent?

If they have the same solution set.

Two linear systems are called equivalent if they have the same solution set. In other words, if any solution that satisfies the first system also satisfies the second system, and vice versa.

To determine if two linear systems are equivalent, we can use methods such as Gaussian elimination or row operations. By applying the same operations to both systems, we can transform them into a simplified form and compare them. If the resulting systems have the same variables and constants, they are equivalent.

For example, consider the following systems:

System 1:

2x + 3y = 7
4x – y = 2

System 2:

x – 2y = 4
-2x + 11y = 23

To determine if they are equivalent, we can perform Gaussian elimination or row operations:

System 1:

[ 2 3 | 7 ]
[ 4 -1 | 2 ]

R2 -> R2 – 2R1

[ 2 3 | 7 ]
[ 0 -7|-12]

R2 -> -1/7 R2

[ 2 3 | 7 ]
[ 0 1 |-12/7]

R1 -> R1 – 3R2

[ 2 0 | 43/7]
[ 0 1 |-12/7]

System 2:

[ 1 -2 | 4 ]
[-2 11 | 23]

R2 -> R2 + 2R1

[ 1 -2 | 4 ]
[ 0 7 |31 ]

R2 -> 1/7 R2

[ 1 -2 | 4 ]
[ 0 1 |31/7]

R1 -> R1 + 2R2

[ 1 0 | 98/7]
[ 0 1 |31/7]

Since both systems have the same variables and constants after simplification, they are equivalent.

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