Understanding the Solution Set of a Linear System | Explained with Examples

The solution set of a linear system involving variables x1, …, xn is a list of numbers (s1,…,sn) that makes each equation in the system a true statement w hen the values s1,…,sn are substituted for x1, …, xn respectively.

A linear system consists of multiple linear equations involving variables x1,

A linear system consists of multiple linear equations involving variables x1, …, xn. The solution set of a linear system is the set of values (s1, …, sn) for the variables x1, …, xn that satisfy all the equations in the system simultaneously.

To determine the solution set, we substitute the values s1, …, sn into the equations and check if each equation holds true. If all the equations are satisfied, then (s1, …, sn) is a solution to the system.

For example, let’s consider a linear system with two equations:

1) 2x + 3y = 10
2) 4x – y = 5

To find the solution set, we substitute values (s1, s2) for (x, y) and check if the equations hold true. Let’s try substituting (1, 2):

1) 2(1) + 3(2) = 10
2) 4(1) – (2) = 5

The first equation becomes 2 + 6 = 10, which is true. The second equation becomes 4 – 2 = 5, which is not true. Hence, (1, 2) is not a solution to the system.

Let’s try another set of values, (3, 1):

1) 2(3) + 3(1) = 10
2) 4(3) – (1) = 5

The first equation becomes 6 + 3 = 10, which is not true. The second equation becomes 12 – 1 = 5, which is true. Hence, (3, 1) is also not a solution to the system.

In this case, the linear system does not have a solution. The solution set would be empty or denoted as { }.

However, if we find specific values of (x, y) that satisfy both equations, then those values will form the solution set. The solution set can be a single point, a line, or even a plane, depending on the complexity of the linear system.

Remember that the solution set of a linear system consists of the values (s1, …, sn) that satisfy all the equations when substituted for the variables (x1, …, xn) in the system.

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