Question

How do you solve the following system of two equations in three variables given
\[x + 2y + z = 4\], \[3x - y - 4z = - 9\]?

Answer 1

Hint: Here in this question, given the system of two linear equations in three variables. We have to find the unknown values that are \[x\],\[y\]and \[z\], solving these equations by using the elimination method. In elimination methods either we add or subtract the equations to find the unknown values of\[x\], \[y\]and \[z\].

Complete step-by-step solution:
Let us consider the equation and we will name it as (1) and (2)
\[x + 2y + z = 4\]----------(1)
\[3x - y - 4z = - 9\]----------(2)
Now we have to solve these two equations to find the unknown
Multiply (1) by 3, then we get
\[3x + 6y + 3z = 12\]
\[3x - y - 4z = - 9\]
Since the coordinates of \[x\] are same and we change the sign by the alternate sign and we simplify to known the unknown value \[y\]in terms of z
\[
   + 3x + 6y + 3z = + 12 \\
 \underline {\mathop + \limits_{( - )} 3x\mathop - \limits_{( + )} y\mathop - \limits_{( + )} 4z = \mathop - \limits_{( + )} 9} \\
  \]
Now we cancel the \[x\] term so we have
\[
   + 3x + 6y + 3z = + 12 \\
  \underline {\mathop + \limits_{( - )} 3x\mathop - \limits_{( + )} y\mathop - \limits_{( + )} 4z = \mathop - \limits_{( + )} 9} \\
    \\
  7y + 7z = 21 \\
 \]
Divide both side by 7, then
\[ \Rightarrow \,\,\,y + z = 3\]
\[\therefore \,\,y = 3 - z\]
We have found the value of \[y\] in terms of z, now we have to find the value of \[x\] . so we will substitute the value \[y\]to any one of the equation (1) or (2) . we will substitute the value of \[y\]to equation (1).
Therefore, we have \[x + 2y + z = 4\]
 \[ \Rightarrow x + 2\left( {3 - z} \right) + z = 4\]
\[ \Rightarrow x + 6 - 2z + z = 4\]
On simplification, we get
\[ \Rightarrow x + 6 - z = 4\]
Subtract 6 on both side
\[ \Rightarrow x + 6 - z - 6 = 4 - 6\]
\[ \Rightarrow x - z = - 2\]
\[\therefore \,\,x = z - 2\]
Hence,
\[\,x = z - 2\] and \[\,y = 3 - z\]
There is no complete solution. If you have 3 variables you need 3 restricting equations to (possibly) find fixed values for your variables.

Note: In this type of question while substituting the term we must be aware of the sign where we change the sign by the alternate sign. In this we have a chance to verify our answers. In the substitution method we have made the one term have the same coefficient such that it will be easy to solve the equation.