In the diagram, MNPQ is a circle with centre O. MNQ = y extordmasculine, NEP = x extordmasculine, NMP = (100−x) extordmasculine and NFP = (5x) extordmasculine. Find the values of x and y.

The angle subtended by an arc at the circumference is half the measure of the arc.
Given $\angle NMP = (100-x)^\circ$, the measure of arc NP is $2 \times (100-x)^\circ$.
Given $\angle MNQ = y^\circ$, the measure of arc MQ is $2 \times y^\circ$.

The angle formed by two chords intersecting inside a circle is half the sum of the measures of the intercepted arcs. $\angle NFP$ is formed by chords NP and MQ intersecting at F. The intercepted arcs are NP and MQ.

Substitute the given values: $\angle NFP = (5x)^\circ$.

Rearrange to form Equation 1:

The angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. $\angle NEP$ is formed by secants NE and ME intersecting at E. The intercepted arcs are NP and MQ.

Substitute the given values: $\angle NEP = x^\circ$.

Rearrange to form Equation 2:

We have the system:

1. $6x - y = 100$
2. $2x + y = 100$

Add Equation (1) and Equation (2):

Substitute $x=25$ into Equation (2):

The values are $x=25, y=50$.