The notion of a system of linear equations

Solving linear equations and inequalities: Systems of linear equations in two unknowns

The notion of a system of linear equations

With a system of linear equations we mean one or more linear equations with one or more unknowns.

A solution of the system of equations is a list of values of the unknowns that, when entered in each equation of the system, makes all equalities true.

Solving a system of equations is the determination of all solutions.

We usually impose a fixed order of the unknowns and write solutions as lists with the values of the variables in the given order.

A typical example is the system

$\left\{\;\begin{aligned} 2x+3y &= 1 \\ 5x+7y&= 3\end{aligned} \right.$ with unknowns $x$ and $y$ , which we also write as

${2 x +3y = 1 \quad \land\quad 5 x +7y =3 }$ Here, $\land$ is the logical "and" operator.

If we choose the order of the unknowns as $x,y$ , then the dyad

$[x=2,y=-1]$ represents a solution. To see that this is indeed a solution, we substitute the values in the equations:

$\left\{\;\begin{aligned} 2\times 2 + 3\times -1 &= 1 \\ 5\times 2 + 7\times -1 &= 3\end{aligned} \right.$ These equalities are true, so $[x=2,y=-1]$ is a solution. We also write this solution in the form

$[x,y]=[2,-1]$ and in the forms

$x=2\quad\text{and}\quad y=-1$ and

$x=2\quad\land\quad y=-1$ Solving a system of equations is the determination of all the solutions. In this case, there are no more solutions.

Mathcentre video

Simultaneous Linear Equations - Animation (2:10)