Question

From the given three equations $x - 3y - 2z = 0$, $5x + 7y - z = 0$ and $ - 7x - 2y + 4z = 0$, calculate the values of $x,y$ and $z$.

Answer 1

Hint: The above question is the example of the simultaneous linear equations. They can be easily solved using certain steps. It can be seen that we have three variables $x,y$ and $z$ . So we have to eliminate one variable from both of them. Now we will be left with two linear equations with two variables. So these equations can be easily solved further and values of $x,y$ and $z$ can be easily calculated.

Complete step-by-step answer:
We are given three equations
  $x - 3y - 2z = 0 - - - - - \left( 1 \right)$, $5x + 7y - z = 0 - - - - - - - \left( 2 \right)$ and$ - 7x - 2y + 4z = 0 - - - - - \left( 3 \right)$
It can be easily observed that each variable has power $1$, so these equations are simultaneous equations. It is obvious that we cannot solve and get the value of each variable by simply adding or subtracting these equations.
The approach we will take is to take $2$ pairs from the above three equations and eliminate one variable in each.
By using the equations $1$ and $2$, we will try to eliminate variable $z$. So, we will subtract $2 \times \left( 2 \right)$ from$\left( 1 \right)$.
 $x - 3y - 2z - 2\left( {5x + 7y - z} \right) = 0$
$x - 3y - 2z - 10x - 14y + 2z = 0$
$ - 9x - 17y = 0$
$9x + 17y = 0$\[ - - - - - - - \left( 4 \right)\]
By using equations $2$ and $3$, here also we will try to eliminate the variable \[z\]. So we will add $4 \times \left( 2 \right)$ and $\left( 3 \right)$
$4\left( {5x + 7y - z} \right) + \left( { - 7x - 2y = 4z} \right) = 0$
$20x + 21y - 4z - 7x - 2y + 4z = 0$
$13x + 19y = 0$$ - - - - - - \left( 5 \right)$
Now we have simultaneous equations in two variables, $\left( 4 \right)$ and$\left( 5 \right)$.
These equations can be solved easily by eliminating one variable
\[9\left( {13x + 19y} \right) - 13\left( {9x + 17y} \right) = 0\]
$117x + 171y - 117x - 221y = 0$
$50y = 0$
So $13x + 19y = 0$
$13x + 19\left( 0 \right) = 0$
$13x = 0$
$x = 0$
And using equation$1$,
 $x - 3y - 2z = 0$
$0 - 0 - 2z = 0$
$z = 0$
The value of the variable $x,y,z$ are all $0$.

Note: At the elimination step, the student must add or subtract the equations such that the variable gets eliminated. We can multiply the whole equation with constants or integers to help in elimination. This makes the problem simpler.