Fifteen triangles and quads have 53 angles. How many triangles and quadrilaterals are there?

Let’s take the number of triangles for x, and for the quadrangles.
1. Based on the condition, we compose two equations:
1) x + y = 15,
2) x 3 + y 4 = 53.
2. Let us express x from the first equation in terms of y:
x + y = 15,
x = 15 – y.
3. Substitute the value of x in the second equation and determine the value of y:
x 3 + y 4 = 53,
(15 – y) 3 + 4 y = 53,
45 – 3 y + 4 y = 53,
45 + y = 53,
y = 53 – 45,
y = 8 – the number of quadrangles.
4. Let’s calculate the value of x:
x = 15 – y = 15 – 8 = 7.
Verification:
x + y = 8 + 7 = 15,
x 3 + y 4 = 7 3 + 8 4 = 21 + 32 = 53.
Answer: 7 triangles, 8 quadrangles.