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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:


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