Solving linear equations and inequalities: Systems of linear equations in two unknowns
Solving systems of equations by the elimination method
The second method to solve a system of two linear equations with two unknowns is the elimination method, in which you try to eliminate the unknown \(x\) (or \(y\) if that is more practical) from the second equation.
We explain the elimination method by example.
Solve the following system of equations: \[\left\{\;\begin{matrix} 2x\!\!\!\!&+&\!\!\!\!5y\!\!\!\!&=&\!\!\!\!9 \\ 3x\!\!\!\!&-&\!\!\!\!4y\!\!\!\!&=&\!\!\!\!2\end{matrix} \right.\]
- We multiply left-hand and right-hand sides of the first equation by 3 and we multiply left-hand and right-hand sides of the second equation by 2. This gives the system \[\left\{\;\begin{matrix} 6x\!\!\!\!&+&\!\!\!\!15y\!\!\!\!&=&\!\!\!\!27 \\ 6x\!\!\!\!&-&\!\!\!\!8y\!\!\!\!&=&\!\!\!\!4\end{matrix} \right.\] We have achieved that coefficient of \(x\) in the two equations are the same.
- If we subtract the first equation from the second, we get \[-23y=-23\] So \[y=1\]
- We find the value for \(x\) by substituting \(y=1\) in the first equation: \[2x+5=9\] So \[x=2\] .
Thus, the solution of the system is \(x=2\) and \(y=1\) .
We will systematise the above elimination process to the reduction method.
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