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)
