The solution of the system of equation with three variables. Given: The syst

Solving system of equation with three variables means finding the value of x, y and z which will make all the three equation TRUE. There are different ways of solving a system of equation, namely substitution method, elimination method and graphical method. Here we can initially use elimination method to eliminate a variable, say z to get system of equation with two variables. When solve the resulting system of equations with two variables using elimination method, we found that the resulting two equations are similar equations. Thereby this system of equations with three variables has infinite number of solutions.
Calculation:
$\begin{array}{|cc|}\hline Description& \text{Solving system of equations:}x-4y+2z=-2;x+2y-2z=-3;x-y=4\\ \text{Step 1: We need to get of rid of fraction by multiplying the Least Common Denominator on both sides of all the three equations to get the new equation (1)(2) and (3). Here in this problem we do not have fractions we can set it straight away.}& x-4y+2z=-2\to 1stequationx+2y-2z=-3\to 2ndequationx-y+0z=4\to 3rdequation\\ \text{Step 2: In an attempt of getting rid of variable z, we need to add 1-st equation and 2-nd equation, to get 4-th equation.}& eqn(2)\Rightarrow x+2y-2z=3eqn(1)\Rightarrow x-4y+2z=-22x-2y=-5\to 4thequation\\ \text{Step 3: Here no need to subtract 3-rd equation from 2-nd equation as third equation itself doesn't contain z}& x-y=4\to 3rdequation\\ \text{Step 4: Subtracting 3-rd equation and 4-th equation, we end up with a contradicting equation it means we have no solutions, that is the graph of system of equation are parallel and never meet. The system equation hence described as inconsistent. Thereby this system of equations has no solution.}& 2x-2y=-5\to 4thequation\Rightarrow x-y=-52x-y=4\Rightarrow 3rdequation(-)(+)(-)0\ne -132\\ \hline\end{array}$
Conclusion: The system of equation with three variables has no solution.