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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.

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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.