Simultaneous Equations (Leaving Cert Mathematics): Revision Notes
Linear Simultaneous Equations in Three Unknowns
A system of three linear equations involves three variables, typically denoted as , and .
Equations of three variables exists on a three-dimensional plane and instead of representing a line, they represent a plane. To solve three equations in three unknowns simultaneously means finding the point of intersection between all three of those planes.
Method 1 : Elimination
The most common method to solve simultaneous equations of three unknowns is to use the elimination method.
- Pick any two equations and eliminate a variable of your choice.
- Pick another pair of equations (different to STEP 1) and eliminate the same variable.
- You'll be left with two equations of two unknown variables, so you can do simultaneous equations of two unknowns.
- After solving both of the variables from STEP 3, substitute them into one of the original equations.
Example
Solve :
Eliminate one variable from any two equations, let's choose equation A and B :
Multiply the top equation by -3
Now eliminate the same variable from another pair of equations :
Multiply by -1 on the top equation :
Not solve for and :
Multiply by -17 :
Finally solve for in one of the original three equations :