Answer
The numbers are $7,4$ and $5$.
Work Step by Step
Let the three numbers be $a,b$ and $c$.
From the question we have
$\Rightarrow a+b+c=16$ ...... (1)
$\Rightarrow 2a+3b+4c=46$ ...... (2)
$\Rightarrow 5a-b=31$ ...... (3)
Multiply the first equation by $4$.
$\Rightarrow 4(a+b+c)=4(16)$
$\Rightarrow 4a+4b+4c=64$ ...... (4)
Subtract equation (2) from equation (4).
$\Rightarrow 4a+4b+4c-(2a+3b+4c)=64-46$
Simplify.
$\Rightarrow 4a+4b+4c-2a-3b-4c=64-46$
Add like terms.
$\Rightarrow 2a+b= 18$ ...... (5)
Add equation (3) and (5).
$\Rightarrow 5a-b+2a+b=31+18$
Simplify.
$\Rightarrow 7a=49$
Divide both sides by $7$.
$\Rightarrow a=7$.
Substitute the value of $a$ into equation (5).
$\Rightarrow 2(7)+b= 18$
Simplify.
$\Rightarrow 14+b= 18$
Subtract $14$ from both sides.
$\Rightarrow 14+b-14= 18-14$
Simplify.
$\Rightarrow b= 4$
Substitute the values of $a$ and $b$ into equation (1).
$\Rightarrow 7+4+c=16$
Add like terms.
$\Rightarrow 11+c=16$
Subtract $11$ from both sides.
$\Rightarrow 11+c-11=16-11$
Simplify.
$\Rightarrow c=5$
