Question

How do you solve the system \[2x-y+3z=2\] , \[x-2y+3z=1\] and \[4x-y+5z=5\] ?

Answer 1

Hint: These types of problems are pretty straight forward and are very easy to solve. For such types of problems, we require a deep understanding of linear equations and simultaneous equations. In such problems, we need to find the value of the three unknown parameters, namely, ‘x’, ‘y’ and ‘z’. First of all we need to multiply the first and the second equation with suitable integers, so that on adding or subtracting these equations, one of the parameters is eliminated. We now choose the second and third equation and eliminate the previous parameter by multiplying both the equations with a suitable integer and then add or subtract accordingly. From the above steps, we form two new equations. Now following the same above steps, we find the value of the two unknown parameters from the two newly formed equations. After that we find the value of the last unknown parameter by putting the value of the other two found out parameters in any of the given equations.

Complete step by step solution:
Now we start off with the solution to the problem by writing the two equations as,
\[2x-y+3z=2\] ---- Equation \[1\]
\[x-2y+3z=1\] ----- Equation \[2\]
\[4x-y+5z=5\] ----- Equation \[3\]
We now multiply the first equation with \[2\] on both the sides and then subtract the second equation from it, we get,
\[2\left( 2x-y+3z \right)-\left( x-2y+3z \right)=2\times 2-1\]
\[\begin{align}
  & 4x-2y+6z-x+2y-3z=2\times 2-1 \\
 & \Rightarrow 3x+3z=3 \\
\end{align}\]
\[\Rightarrow x+z=1\] ---- Equation \[4\]
We multiply the third equation with \[2\] on both the sides and then subtract the second equation from it we get,
\[2\left( 4x-y+5z \right)-\left( x-2y+3z \right)=5\times 2-1\]
\[\begin{align}
  & 2\left( 4x-y+5z \right)-\left( x-2y+3z \right)=5\times 2-1 \\
 & \Rightarrow 8x-2y+10z-x+2y-3z=9 \\
 & \Rightarrow 7x+7z=9 \\
\end{align}\]
\[7x+7z=9\] ---- Equation \[5\]
Now, we see the two equations formed, i.e. equation $4$ and equation $5$ , we see that these two equations are contradictory in nature, because if one exists then the other cannot. Thus there are no solutions to this problem.

Note: For these types of problems we need to study the chapter of theory of linear and simultaneous equations very well. This type of problem can also be solved in another way, i.e. by using the theory of graphs. These equations are nothing but the equations of planes in a three dimensional coordinate system. Plotting these three planes and finding the resultant intersection point gives us the solution of the three simultaneous equations of ‘x’, ‘y’ and ‘z’.