To solve for the magnitude of vector c, defined as c = 5a - 2b, we start with the given vectors: a = (3, 1) and b = (-4, 9). The first step is to calculate 5a and 2b.

Calculating 5a involves multiplying each component of vector a by 5:

5a = 5 * (3, 1) = (5 * 3, 5 * 1) = (15, 5).

Next, we calculate 2b by multiplying each component of vector b by 2:

2b = 2 * (-4, 9) = (2 * -4, 2 * 9) = (-8, 18).

Now, we can find vector c by subtracting 2b from 5a:

c = 5a - 2b = (15, 5) - (-8, 18) = (15 + 8, 5 - 18) = (23, -13).

With vector c determined as (23, -13), we can now find its magnitude using the Pythagorean theorem:

|c| = √(x_c² + y_c²), where x_c = 23 and y_c = -13.

Calculating the squares:

23² = 529 and (-13)² = 169.

Adding these results gives:

529 + 169 = 698.

Thus, the magnitude of vector c is:

|c| = √698.

This process illustrates how to handle multiple vector operations and find the magnitude of a resultant vector effectively.