Solving linear equations and inequalities: Systems of linear equations in two unknowns
Solving systems of equations by Gaussian elimination
The elimination method can be systematised to what is known as Gaussian elimination. The strategy is to re-edit the equations; Think of a multiplication of all terms in the same equation with a number, and the subtraction of one equation from the other. This process is row reduction and therefore we call this approach called the method of row reduction or simply row reduction.
Row reduction
1. Make sure that \(x\) occurs in the first equation. If that is not the case, then we swap the two equations of place, so that \(x\) does occur in the first equation.
2. Replace the second equation by the difference of this equation and a suitably selected multiple of the first equation, so that \(x\) is no longer present in the second equation.
3. Replace the first equation by the difference of this equation with an appropriate multiple of the second equation so that \(y\) is no longer present in the first equation.
4. By multiplying the first and second equation with a suitable number, we ensure that the left-hand side becomes \(x\) respectively \(y\), so that the solution appears.
We exemplify this method via the earlier example of a system of two linear equations in two unknowns.
Solve the following system of equations: \[\left\{\;\begin{aligned}2x + 3y &= 1 \\ 3x + 7y &= -1\end{aligned} \right.\]
1. The unknown \(x\) is already present in the upper equation. We do not need to change the equations.
2. We subtract \(\tfrac{3}{2}\) times the first equation of the second. This eliminates the term with \(x\) and creates the equation \[7y-\tfrac{3}{2}\cdot 3y = -1-\tfrac{3}{2}\] So \(\tfrac{5}{2}y=-\tfrac{5}{2}\) and thus \(y=-1\). The equivalent system of equations is \[\left\{\;\begin{aligned} 2x+3y &= 1 \\\phantom{2x+3}y &= -1\end{aligned} \right.\]
3. We subtract now three times the second equation from the first one (so that the term with \(y\) disappears) and get the system \[\left\{\;\begin{aligned} 2x &= 4 \\ \phantom{2}y &= -1\end{aligned} \right.\]
4. We divide the first equation by \(2\) to get the system \[\left\{\;\begin{aligned} x &= 2 \\ y &= -1\end{aligned} \right.\] Herewith the solution of the system of equations has been found.
Mathcentre videos
Solving a System of Linear Equations (34:21)
\(\phantom{x}\)
Simultaneous Linear Equations - Animation (2:10)
