The number of edges can be calculated from the number of vertices.
The variable N is used to always represent the number of vertices.
So, we represent the edges as:
[tex]E \to Edges[/tex]
(a) The value of N for 105 edges
The relationship between N and E is:
[tex]E = \frac{N \times (N - 1)}{2}[/tex]
Substitute 105 for E
[tex]105 = \frac{N \times (N - 1)}{2}[/tex]
Multiply through by 2
[tex]210 = N \times (N - 1)[/tex]
[tex]210 = N^2 - N[/tex]
Rewrite as:
[tex]N^2 - N - 210 = 0[/tex]
Expand
[tex]N^2 +14N - 15N - 210 = 0[/tex]
Factorize
[tex]N(N +14) - 15(N + 14) = 0[/tex]
Factor out N + 14
[tex](N - 15) (N + 14) = 0[/tex]
Solve for N
[tex]N = 15[/tex] or [tex]N = -14[/tex]
The number of vertices (N) cannot be negative. So:
[tex]N = 15[/tex]
(b) The value of N for 19900 edges
We have:
[tex]E = \frac{N \times (N - 1)}{2}[/tex]
Substitute 19900 for E
[tex]19900 = \frac{N \times (N - 1)}{2}[/tex]
Multiply through by 2
[tex]39800 = N \times (N - 1)[/tex]
[tex]39800= N^2 - N[/tex]
Rewrite as:
[tex]N^2 - N - 39800= 0[/tex]
Expand
[tex]N^2 +199N - 200N - 39800= 0[/tex]
Factorize
[tex]N(N +199) - 200(N + 199) = 0[/tex]
Factor out N + 199
[tex](N + 199) (N - 200) = 0[/tex]
Solve for N
[tex]N = 200[/tex] or [tex]N = -199[/tex]
The number of vertices (N) cannot be negative. So:
[tex]N = 200[/tex]
Hence, there are 200 vertices for 19900 edges
Read more about vertices and edges at:
https://brainly.com/question/22118318
