Understanding Solution Sets and Equivalent Systems of Equations in Mathematics

What is a solution set, and how does it relate to two systems being equivalent?

In mathematics, a solution set refers to the set of values or elements, which satisfy a given equation or system of equations

In mathematics, a solution set refers to the set of values or elements, which satisfy a given equation or system of equations. It is the collection of all possible solutions that satisfy the given conditions. For example, in a simple linear equation like 2x + 3 = 7, the solution set would contain a single element, which is x = 2. In more complex equations or systems of equations, the solution set might include multiple solutions or be empty.

Now, when we talk about two systems being equivalent, it means that they have the same solution set. In other words, for two systems of equations to be equivalent, they must have the exact same solutions. This implies that every solution for one system is also a solution for the other system, and vice versa. It is important to note that the two systems might have different forms or representations, but they are equivalent as long as they yield the same solutions.

To illustrate this, consider the following example:

System 1:
2x + 3y = 7
5x + y = 1

System 2:
x = 2 – (3y/2)
y = -3x + 1/5

At first glance, these two systems might appear different due to their different forms. However, if we solve both systems, we will find that the solution sets for both systems are the same. This indicates that System 1 and System 2 are equivalent.

So, to summarize, a solution set represents the set of values that satisfy a given equation or system of equations. Two systems are considered equivalent if they have the same solution set, meaning they produce the identical solutions.

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