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I have created two equations based on this statement, let x be the number of vertices with degree 1, y be number of vertices with degree 2, z be number of vertices with degree 3.

$x+2y+3z=20$ from the handshake theorem

and $x+y+z=7$

But am not sure where to proceed. Can anyone help me out. Thanks.

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  • $\begingroup$ Put it as a matrix equation, $\begin{bmatrix} 1&2&3\\1&1&1 \end{bmatrix} \begin{bmatrix} x\\y\\z \end{bmatrix} = \begin{bmatrix} 20\\7 \end{bmatrix}$. Row reduce. You want positive integer solutions. $\endgroup$ – Morgan Rodgers Jun 11 '18 at 6:42
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We have $$ x+2y+3z = 20 \tag1 $$$$ x+y+z = 7 \tag2 . $$ Multiplying $(2)$ by $3$ and subtracting $(1)$ from it we get $$ 2x + y = 1. $$ Since $x,y$ are integers we conclude that $x=0$, $y=1$ and so $z = 6$.

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