Question

How do you solve the system $x-y+z=3$ , $2x+y+z=8$ and $3x+y-z=1$ ?

Answer 1

Hint: We solve the given problem on a linear system using substitution method. First, we find the value of $z$ in terms of $x\text{ and }y$ . Putting the value of $z$in the other two equations we get two equations containing only $x\text{ and }y$ which can be solved by substitution method. Again, putting the value of $x\text{ and }y$ in any of the equations we get the value of $z$ .

Complete step by step solution:
The equations we have are
$x-y+z=3....\left( 1 \right)$
$2x+y+z=8....\left( 2 \right)$
$3x+y-z=1....\left( 3 \right)$
We take equation $\left( 1 \right)$ and add $y$ then subtract $x$ from both the sides as
$\Rightarrow z=3+y-x$
Putting the above value of $z$ in equation $\left( 2 \right)$ as
$\Rightarrow 2x+y+\left( 3+y-x \right)=8$
Simplifying the above equation, we get
$\Rightarrow x+2y=5....\left( 4 \right)$
Putting the value of $z$ in equation $\left( 3 \right)$ as shown below
$\Rightarrow 3x+y-\left( 3+y-x \right)=1$
$\Rightarrow 4x-3=1$
Simplifying the above equation, we get
$\Rightarrow 4x=4$
Further dividing both the sides of the above equation by $4$ we get
$\Rightarrow x=\dfrac{4}{4}$
$\Rightarrow x=1$
Again, substituting the value of $x$ in equation $\left( 4 \right)$ we get
$\Rightarrow 1+2y=5$
Further subtracting $1$ from both the sides of the above equation, we get
$\Rightarrow 2y=4$
Now, we divide both the sides of the above equation by $2$ as shown below
$\Rightarrow y=\dfrac{4}{2}$
$\Rightarrow y=2$
Now, we put $x=1$ and $y=2$ in equation $\left( 1 \right)$ as shown below
$\Rightarrow 1-2+z=3$
Further simplifying the above equation, we get
$\Rightarrow z=4$

Therefore, we conclude that the solutions of the given equations are $x=1$ , $y=2$ and $z=4$ .

Note: We can also solve the given equations using other methods. We could use the Matrix method or we could use an elimination method which will give us two linear equations which can be solved as we have done in our solution. Also, while using the substitution method we have to keep in mind that we must properly substitute the expression in the equation so that mistakes can be avoided.