How To Solve A System Of Equations With 3 Variables?
To solve a system of three equations with three variables you can use substitution, linear combinations, or any number of matrix applications including Cramer's method and Gaussian elimination.
Suppose we are asked to solve a linear system of three equations in three variables. If a unique solution exists it will be an ordered three-tuple (e.g. (x,y,z) ) which represents a point in Euclidean 3-space. The system will be said to be consistent and independent.
If the solution is not unique we can present the solution as an ordered three-tuple with each element written in terms of one variable. E.g. the solution might appear as (x, x+2, -2x). The system is said to be consistent and dependent.
It is also possible that the system has no solution in which case it is said to be inconsistent.
Graphically a system that is consistent and independent is the point of intersection of three planes (like the corner of a room.) If the system is consistent and dependent the solution is the intersection of three planes in a line (much like the spine of a book.)
To solve we can use techniques we already know from systems of two equations in two unknowns.
Example: Solve the following system:
A. We can use substitution. Choose an equation and a variable to solve for. In this case it would be simple to choose to solve the first equation for x.
`x=-2y-3z+2` Now we substitute this expression for x in the remaining equations.
`-2(-2y-3z+2)-3y+2z=2` Simplifying we get:
Now we have two equations in two unknowns and we have many techniques to solve this system. Solving the second equation for y and substituting we get
`-11(-8z+6)-13z=9 ==> 75z=75 ==> z=1`
Now we can back substitute to get y then x:
`y+8=6 ==> y=-2`
The solution is (3,-2,1) which we would check in each of the three original equations.
B. We can also use linear combinations:
Choose to eliminate x; if the equations are marked i, ii, iii then take 3i-ii to get 11y+13z=-9
Then 2i+iii to get y+8z=6
Now we have the system:
Eliminating y we get 75z=75 as above.
There are many techniques using matrices (Cramer's method, Gaussian elimination, matrix multiplication etc...) as well as some linear algebra techniques with basis vectors.