What are the solution sets for equations with 3 variables?
Set of Intersections of the Planes:1.) Line 2.) Point 3.) Nothing
The solution sets for equations with 3 variables are typically expressed as ordered triples {x, y, z} that satisfy all the given equations simultaneously.
For example, consider the system of equations:
x + 2y – z = 5
2x – y + 3z = 10
3x + 2y + z = 2
To solve this system, we can use techniques such as elimination, substitution, or matrices. Here’s one possible approach using elimination:
Step 1: Multiply the first equation by 2 and subtract it from the second equation to eliminate y:
2x – y + 3z = 10
-3x + 5y – 7z = 0
Step 2: Multiply the first equation by 3 and add it to the third equation to eliminate x:
x + 2y – z = 5
7y + 2z = -13
Step 3: Solve for y and z by back-substitution:
7y + 2z = -13
y = (-13 – 2z)/7
x + 2y – z = 5
x + 2((-13 – 2z)/7) – z = 5
x = (39 – 17z)/7
Therefore, the solution set of this system is given by { (39-17z)/7, (-13-2z)/7, z }, where z is any real number.
Alternatively, we can use matrices to solve this system. We write the augmented matrix [A|B], where A is the coefficient matrix and B is the constant matrix:
[ 1 2 -1 | 5 ]
[ 2 -1 3 | 10 ]
[ 3 2 1 | 2 ]
Then we perform row operations to reduce the matrix to row-echelon form:
[ 1 2 -1 | 5 ]
[ 0 -5 5 | 0 ]
[ 0 0 7 | -13 ]
Finally, we use back-substitution to solve for the variables:
x = (39-17z)/7
y = (-13-2z)/7
z = -13/7
Therefore, the solution set of this system is also given by { (39-17z)/7, (-13-2z)/7, -13/7 }, where z is any real number.
More Answers:
Row Equivalence In Linear Equations: Elementary Row Operations ExplainedMatrix Size: Explaining M*N Notation For Rows And Columns
Consistent And Inconsistent Systems Of Equations In Math