Final answer:
The system of equations might be consistent with a unique solution but cannot be definitively categorized without further simplification or use of methods such as matrix analysis to establish whether the remaining two equations are independent.
Step-by-step explanation:
To determine the nature of the system of linear equations, one can use several methods such as graphing, substitution, elimination, or matrix analysis. The given system of equations is:
- 2x₁ − 2x₃ = 10
- 2x₁ + 3x₂ + x₃ = 20
- −x₁ + x₂ + 2x₃ = 5
To solve for x₁, x₂, and x₃, let's apply the elimination method:
- First, add the first and third equations to eliminate x₃:
2x₁ − 2x₃ + (-x₁ + x₂ + 2x₃) = 10 + 5
x₁ + x₂ = 15 -- (Equation 4)
- Now, subtract the first equation from the second one:
2x₁ + 3x₂ + x₃ - (2x₁ − 2x₃) = 20 - 10
3x₂ + 3x₃ = 10 -- (Equation 5)
Now we have a simpler system with just two variables, x₁ and x₂:
x₁ + x₂ = 15
3x₂ + 3x₃ = 10
By analyzing the simplified system, we can see if the equations are multiples of each other, proceed to find one solution, or determine if there's no solution. Without additional manipulation to reduce Equations 4 and 5 further, we cannot definitively determine if the system is consistent with a unique solution, consistent with infinitely many solutions, or inconsistent. However, since we started with three equations and three unknowns and were able to reduce the system to two equations with two unknowns, there is a possibility that this system might have a unique solution if the two remaining equations are independent.