Simultaneous Equations with 3 Variables
To be able to solve simultaneous equations with 3 unknowns; we will need 3 equations.
We can use similar methods to when we solve with 2 unknowns; elimination or substitution.
How do I solve simultaneous equations with 3 unknowns using elimination?
- The general approach is to eliminate one of the variables from two equations; so that we are left with 2 equations in terms of 2 unknowns
- Once we have 2 equations with 2 unknowns, we can use the methods we already know to solve them, and then substitute the values back in to find the 3rd unknown
- It is very important to label the equations as there are lots to keep track of
- Take for example the following simultaneous equations
- We can eliminate by adding equations and
- This is equation
- We can form another equation containing only and , by eliminating , by taking equation and adding on 2 times equation
- This is equation
- Equations and can then be solved using the previously covered methods to find and
- Substitute these values into any of the original three equations to find
- As a check, we should substitute the found values for into each of the original equations to make sure they all work
How do I solve simultaneous equations with 3 unknowns using substitution?
- The general approach is the same; eliminate one of the variables from two equations; so that we are left with 2 equations in terms of 2 unknowns
- With elimination we did this by adding or subtracting the equations (or multiples of the equations)
- With substitution we do it by rearranging one of the equations to make or the subject, and then substituting this into the other two equations
- Once we have 2 equations with 2 unknowns, we can use the methods we already know to solve them, and then substitute the values back in to find the 3rd unknown
- It is very important to label the equations as there are lots to keep track of
- Take for example the following simultaneous equations
- We can rearrange equation to make the subject
- Substitute into equation
- Simplify into the form
- , call this equation
- Substitute into equation
- Simplify into the form
- , call this equation
- Equations and can then be solved using the previously covered methods to find and
- Substitute these values into any of the original three equations to find
- As a check, we should substitute the found values for into each of the original equations to make sure they all work
Exam Tip
- Label each equation you use and write down exactly what you are doing in each step
- e.g. or
- Always check your final solutions work for all three original equations
- You can leave the exam knowing that you got the answer correct!
Worked example
Solve the simultaneous equations.
Method 1: Elimination
Number the equations.
Eliminate by adding equations (1) and (2).
(1) + (2):
Find a second equation in and by subtracting 2 times equation (1) from equation (3).
(3) - 2(1):
Solve equations (4) and (5) using your preferred method of solving a pair of linear simultaneous equations.
(4) -5(5):
Substitute into equation (4) or (5) and solve to find .
Substitute and back into one of equation (1), (2) or (3) and solve to find
Substitute the three solutions into the other two equations to check that they are correct.
Method 2: Substitution
Number the equations.
Rearrange equation (1) to make the subject.
(1):
Substitute into equation (2) and simplify.
(2):
Substitute into equation (3) and simplify.
(3):
Solve equations (4) and (5) using your preferred method of solving a pair of linear simultaneous equations.
(4) -5(5):
Substitute into equation (4) or (5) and solve to find .
Substitute and back into one of equation (1), (2) or (3) and solve to find
Substitute the three solutions into the other two equations to check that they are correct.