Linear Simultaneous Equations in Three Unknowns

A system of three linear equations involves three variables, typically denoted as $x$ , $y$ and $z$ .

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.

infoNote

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

infoNote

Solve :

7x + y + 5z = 27 \quad \text{(Equation A)} \\ 4x + 3y + 5z = 21 \quad \text{(Equation B)} \\ 6x + y + 2z = 9 \quad \text{(Equation C)}

Eliminate one variable from any two equations, let's choose equation A and B :

7x + y + 5z = 27 \\ 4x+3y+5z=21

Multiply the top equation by -3

-21x - 3y - 15z = -81 \\ \underline{4x+3y+5z=21}

-17x-10z=-60

Now eliminate the same variable from another pair of equations :

7x + y + 5z = 27 \\ 6x+y+2z=9

Multiply by -1 on the top equation :

-7x - y - 5z = -27 \\ \underline{6x+y+2z=9} \\ -x-3z=-18

Not solve for $x$ and $z$ :

-17x-10z=-60 \\ -x-3z=-18

Multiply by -17 :

-17x-10z=-60 \\ \underline{17x+51z=306} \\ 41z=246

z=:success[6]

-x-3(6)=-18

x=:success[0]

Finally solve for $y$ in one of the original three equations :

7x + y + 5z = 27

7(0) + y + 5(6) = 27

y=:success[-3]

Linear Simultaneous Equations in Three Unknowns Simplified Revision Notes for Leaving Cert Mathematics

Linear Simultaneous Equations in Three Unknowns

Method 1 : Elimination

500K+ Students Use These Powerful Tools to Master Linear Simultaneous Equations in Three Unknowns For their Leaving Cert Exams.

Other Revision Notes related to Linear Simultaneous Equations in Three Unknowns you should explore