let

x = the cost of 1 cup

y = the cost of 1 plate

7 cups and 8 plates cost $9.40:

7x + 8y = 9.40

8 cups and 7 plates cost $9.65:

8x + 7y = 9.65

if we add the two equation by adding the left and the right sides we get:

7x + 8y + 8x + 7y = 9.40 + 9.65

15(x + y) = 19.05 divide both sides by 15

x + y = 1.27

the cost of a cup and a plate is $1.27.