Mastering Linear Algebra: Understanding Equivalent Linear Systems and their transformation through Row Operations

equivalent linear systems

linear systems with the same solution set

Equivalent linear systems refer to a concept in linear algebra where two systems of linear equations are said to be equivalent if they have the same solution set. In other words, if two linear systems have the same solution set, they are said to be equivalent.

To demonstrate this concept, consider two linear systems as follows:

System 1:
2x – y = 3
3x + 4y = 2

System 2:
4x – 2y = 6
6x + 8y = 4

To determine if these systems are equivalent, we can use several methods. One such method is to use row operations to transform one system into the other. Row operations involve adding, subtracting, or scaling the rows of a matrix to transform it into a simpler or more useful form.

Using row operations, we can transform System 1 into System 2 as follows:

– Multiply the first row by 2 to get:
4x – 2y = 6
3x + 4y = 2

– Subtract 2 times the first row from the second row to get:
4x – 2y = 6
2x + 8y = -10

– Divide the second row by 2 to get:
4x – 2y = 6
x + 4y = -5/2

Looking at these two systems, we can see that they have the same solution set, which is (x= -2, y= -1/2). Therefore, they are equivalent linear systems.

In summary, equivalent linear systems are systems of linear equations that have the same solution set. They can be determined by using row operations to transform one system into another or by analyzing their coefficients and constants.

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