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Given, arc QR is congruent to arc LN and arc OP is congruent to arc VW.

And the expressions for each arc in the diagram also given as:

Arc QR = 2x - y, arc LN = 11 , arc OP= 10 and arc VW=5x+y.

Hence, we will get the system of equations as following:

Arc QR = Arc LN

2x - y = 11 ...(1)

Arc OP = Arc VW

5x + y = 10 ...(2)

Next step is to add the two equation to eliminate y so that we can solve the equations for x. Therefore,

2x+5x = 11 + 10

7x = 21

[tex] \frac{7x}{7} =\frac{21}{7} [/tex] Divide each sides by 7.

So, x= 3

Now plug in x=3 in equation (2) to get the value of y.

5(3) + y = 10

15 + y =10

15 + y - 15 = 10 - 15 Subtracting 15 from each sides.

y = -5

So, x=3 and y =-5

Answer:  The required values are

x = 3  and  y = 5.

Step-by-step explanation:  Given that the circles M and K are congruent. Also, QR is congruent to LN and OP is congruent to VW.

We are to find the values of x and y.

Since QR is congruent to LN, so their lengths must be equal.

So, we have

x+2 = y                                                                    (i)

Similarly, we can see that the lengths of OP and VW are equal.

That is,

   2x - 1 = y

⇒ 2x - 1 = x + 2         [Using equation (i)]

⇒ 2x-x=2+1

⇒ x = 3.

And, from equation (i), we get

y = x + 2 = 3 + 2 = 5.

Thus, the required values are

x = 3  and  y = 5.

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