# Further Maths - Solving Systems of Equations Using Triangle Method

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/solving-systems-of-equations-triangle/ · Updated: 2025-05-14 · Tags: further-maths, latex-block-alt, school

##### Other than matrices, how can you solve a system of three equations??
Using the triangle method.

##### How does the triangle method of solving a system of three equations work??
Using back substitution.

##### What are the steps involved in the triangle method??
* Add/subtract two equations of three variables to form an equation of two variables
* Do this for another equation, forming an equation with the same two variables
* Solve this using simultaneous equations
* Substitute the new values and solve for the final variable

##### Why is it called the "triangle" method of solving equations??
You work your way from 3 to 1 variables, so it kind of looks like an upside-down pyramid.

##### If you have the three equations $$2x-6y+4z = 32 \\ 3x + 2y -9z = -49 \\ -2x + 4y + z = -3$$, what two equations could you initially pair??
Add equation 1 and equation 3 together.

$$
2x - 6y + 4z = 32
-2x + 4y + z = -3
$$

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