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# FM Linear Equation

## 1.3 Simultaneous Linear Equations

### Systems of Simultaneous Linear Equations

• To solve all unknown values in a set of simultaneous linear equations, the number of unknowns must be the same as the number of unique equations in the set.

### Number of Solutions for Simultaneous Linear Equations

For two simultaneous linear equations

y=m_{1} x+c_{1}

y=m_{1} x+c_{1}

The number of solutions is defined by the gradient and y-intercept of the two equations.

Unique Solution (m_{1} \neq m_{2})

• There is a single value for each variable that satisfies every equation.

No Solutions (m_{1} = m_{2}, and c_{1} \neq c_{2})

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## 2.1 Simple Matrix Recurrence Relations

### The Simple Matrix Recurrence Relation Formula

• The simplest type of matrix recurrence relation formula we will analyse in Further Maths models a system where the next “state”; S_{n+1}, can be reached by multiplying the current state; S_{n}, by a transition matrix; T, in the form:

S_{n+1}=T S_{n}

• The state matrices; S_{n} (where n is a positive whole number representing the state of the system), are column matrix listing the value of each of the system’s variables in the corresponding state.
• The transition matrix; T, is a square matrix.
• As with a linear recurrence relation, it is important to state the initial state of a system; S_{0}.
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## 1.8 Matrix Applications: Solving Systems of Equations

### Representing Systems of Equations in Matrix Form

• Matrices provide a concise way of representing systems of linear equations (i.e. multiple related linear equations). The equations are represented in 3 matrices:
• A square coefficient matrix, generally denoted by a capital A, where each column lists the coefficients of a corresponding variable, while each row corresponds to a different equation.
• A variable column matrix which lists the variables. It is denoted by a capital X. This is multiplied by the coefficient matrix to form the left-hand side of the matrix equation.
• A column matrix of constants, which lists the constant in each equation. It is denoted by a lower-case b. This forms the right-hand side of the matrix equation.
• Equation for a system of linear equations in matrix form:

AX=b

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