Answer
The numbers are $-1,2$ and $3$.
Work Step by Step
Let the three numbers be $a,b$ and $c$.
From the question we have
$\Rightarrow 3a+b+2c=5$ ...... (1)
$\Rightarrow (a+3c)-3b=2$
Rearrange.
$\Rightarrow a-3b+3c=2$...... (2)
$\Rightarrow (2a+3b)-c=1$
Rearrange.
$\Rightarrow 2a+3b-c=1$ ...... (3)
Multiply equation (3) by $2$.
$\Rightarrow 2(2a+3b-c)=2(1)$
$\Rightarrow 4a+6b-2c=2$...... (4)
Add equation (1) and (4).
$\Rightarrow 3a+b+2c+4a+6b-2c=5+2$
$\Rightarrow 7a+7b=7$
Divide the equation by $7$.
$\Rightarrow a+b=1$...... (5)
Multiply equation (5) by $-6$
$\Rightarrow -6a-6b=-6$...... (6).
Multiply equation (3) by $2$.
$\Rightarrow 3(2a+3b-c)=3(1)$
$\Rightarrow 6a+9b-3c=3$...... (7)
Add equation (2) and (6).
$\Rightarrow a-3b+3c+6a+9b-3c=2+3$
$\Rightarrow 7a+6b=5$...... (8).
Add equation (6) and (8).
$\Rightarrow -6a-6b+7a+6b=-6+5$
Simplify.
$\Rightarrow a=-1$
Substitute the value of $a$ into equation (5).
$\Rightarrow -1+b=1$
Add $1$ to both sides.
$\Rightarrow -1+b+1=1+1$
Simplify.
$\Rightarrow b=2$
Substitute the values of $a$ and $b$ into equation (1).
$\Rightarrow 3(-1)+2+2c=5$
Simplify.
$\Rightarrow -3+2+2c=5$
$\Rightarrow -1+2c=5$
Add $1$ to both sides.
$\Rightarrow -1+2c+1=5+1$
Simplify.
$\Rightarrow 2c=6$
Divide both sides by $2$
$\Rightarrow c=3$
