# Strictly triangular systems of linear equations

## Prerequisites

### Cartesian Coordinates Show

### Gradient of a straight line Show

### Equation of a straight line Show

### Systems of linear equations Show

### Coefficient and augmented matrices Show

### Strictly triangular systems of linear equations

#### Introduction

- A system of linear equations is strictly triangular if:
- It is
**n** x **n**, i.e. the number of equations is equal to the number of variables. - For equation
**k**, the coefficients for **x**_{1} to **x**_{k - 1} are 0. - For equation
**k**, the coefficient for **x**_{k} is not 0. - For
**k** in 1, ..., **n**

- A 3x3 strictly triangular system has the form:
- Or as an augmented matrix:

#### Solving strictly triangular systems

- Strictly triangular systems are very easy to solve.
- For example:
- From the 3rd equation, we know that:
- So we can substitute this into the second equation:
- And finally substitute
**x**_{2} and **x**_{3} into our first equation: - This method of solving strictly triangular systems is called back-substitution.

#### Converting systems to be strictly triangular

- If a system of equations has exactly one solution then it can be converted to a strictly triangular system.
- For example, given the system:
- First, we convert it into an augmented martrix:
- We start by eliminating the coefficients of
**x** for all but the first row:- To do this, we subtract multiples of the 1st row from the other rows.
- The row that is used for elimination is called the
**pivotal row**. - We subtract the 1st row from the 2nd:
- And then we subtract 3 times the 1st row from the 3rd row:

- Finally we eliminate the coefficients of
**y** from the 3rd row:- To do this, we use the 2nd row as our pivotal row.
- We subtract 3 times the 2nd row from the 3rd row:

#### Choosing a pivotal row

- Sometimes a system will be given like:
- In this case the first row can't be used as a pivotal row to eliminate the coefficients in the first column.
- Instead, it is necessary to switch round the first two rows.