Question

Solve the following systems of equations:
$
  x - y + z = 4 \\
  x - 2y - 2z = 9 \\
  2x + y + 3z = 1 \\
 $

Answer 1

Hint – First mark the given equations in order of (1), (2) and (3) in the same sequence as given. First subtract equation (2) from equation (1), this will give the relation between y and z, then subtract twice equation (1) from equation (3), this again gives relation between y and z. Solve them to get the value of y and z, then substitute these values back in equation (1) to get the answer.

Complete step by step answer:
Given system of equations:
$
  x - y + z = 4.......................\left( 1 \right) \\
  x - 2y - 2z = 9...................\left( 2 \right) \\
  2x + y + 3z = 1....................\left( 3 \right) \\
$
Now subtract equation (2) from equation (1) we have,
$ \Rightarrow x - y + z - \left( {x - 2y - 2z} \right) = 4 - 9$
Now simplify the above equation we have,
$ \Rightarrow y + 3z = - 5$...................... (4)
Now subtract twice of equation (1) from equation (3) we have,
$ \Rightarrow 2x + y + 3z - 2\left( {x - y + z} \right) = 1 - 2\left( 4 \right)$
$ \Rightarrow 3y + z = - 7$
Now multiply by 3 in this equation we have,
$ \Rightarrow 9y + 3z = - 21$................... (5)
Now subtract equation (4) from equation (5) we have,
$ \Rightarrow 9y + 3z - y - 3z = - 21 - \left( { - 5} \right)$
$ \Rightarrow 8y = - 21 + 5 = - 16$
Now divide by 8 we have,
$ \Rightarrow y = \dfrac{{ - 16}}{8} = - 2$
Now from equation (4) we have,
$ \Rightarrow \left( { - 2} \right) + 3z = - 5$
Now simplify the above equation we have,
$ \Rightarrow 3z = - 5 + 2 = - 3$
Now divide by 3 we have,
$ \Rightarrow z = \dfrac{{ - 3}}{3} = - 1$
Now substitute the value of y and z in equation (1) we have,
$ \Rightarrow x - \left( { - 2} \right) + \left( { - 1} \right) = 4$
$ \Rightarrow x = 4 + 1 - 2 = 3$
So the required solution of the given system of equation is
$ \Rightarrow \left( {x,y,z} \right) = \left( {3, - 2, - 1} \right)$
So this is the required answer.

Note – These type of problems involving three equation of three variables is based upon this simple fact that first manipulate the equations to get the value of two unknowns by forming two linear equations in two variables, then solve them either via substitution or elimination, then use these two known values to get the third variable.