What are the solution sets for equations with 3 variables?
Set of Intersections of the Planes:1.) Line 2.) Point 3.) Nothing
The solution set for equations with 3 variables can take different forms, depending on the type of equations and the constraints. Here are some general categories:
1. No solution: If the equations are inconsistent, i.e., they have no common solution, then the solution set is empty. This may happen, for example, if one of the equations is x + y + z = 2 and another is x + y + z = 5, which cannot be satisfied simultaneously.
2. Unique solution: If the equations are independent and linear, i.e., they have a unique solution, then the solution set is a single point in 3D space. This point can be found by solving the system of equations using row reduction or elimination. For example, the system
x + y + z = 4
2x – y + z = 1
3x + y – z = 7
can be solved by augmented matrix as:
[ 1 1 1 | 4 ]
[ 2 -1 1 | 1 ]
[ 3 1 -1 | 7 ]
and then using elementary row operations to reduce it to:
[ 1 0 0 | 2 ]
[ 0 1 0 |-1 ]
[ 0 0 1 | 3 ]
Therefore, the unique solution is (x, y, z) = (2, -1, 3).
3. Infinitely many solutions: If the equations are dependent or nonlinear, i.e., they have an infinite number of solutions, then the solution set is a subspace of 3D space. To find the solution set, we need to first find the set of parameters that can vary freely without changing the solutions, called the free variables. Then, we can express the dependent variables in terms of the free variables using the equations. For example, the system
x + y + z = 1
x – y + z = 2
2x + y – z = 0
can be solved by augmented matrix as:
[ 1 1 1 | 1 ]
[ 1 -1 1 | 2 ]
[ 2 1 -1 | 0 ]
Using row operations, we can eliminate the third column:
[ 1 0 -2 | 1 ]
[ 0 1 -1 | 1 ]
[ 0 0 0 | 0 ]
Therefore, z is a free variable, and x and y can be expressed in terms of z as:
x = 1 + 2z
y = 1 + z
Thus, the solution set is a line in 3D space, parametrized by z. For any value of z, we get a solution (x, y, z) on the line.
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