Sheila wants to deposit $2,145 into savings accounts at three different banks: bank of the us, 7/4 bank, and catch bank. she will deposit five times as much into catch bank as 7/4 bank because they offer a higher interest rate. she also expects the bank of the us deposit to be 30% of the total of the other two deposits. how much will sheila deposit into the bank of the us savings account?

Let x = deposit for Bank of US; y = deposit for 7/4 Bank and z = deposit for Catch Bank. Since we have 3 unknowns, we have to formulate 3 independent equations from the problem statement.

She will deposit five times as much into Catch Bank as 7/4 Bank => z = 5y
Bank of the US deposit to be 30% of the total of the other two deposits =>
x = 0.3(y+z)
Total deposit => $2145 = x + y +z

There, we have formulated three. Let's assign these equations:
Eq. 1: z = 5y
Eq. 2: x = 0.3(y+z)
Eq. 3: $2145 = x + y +z

Substitute Eq 1 to Eq 2, we get an equation for x in terms of y. This would be Eq. 4. Then, we substitute Eq 4 to Eq 3, to solve for y. 

Eq. 1 to Eq. 2:
x = 0.3 (y + 5y) = 0.3(6y)
x = 1.8y => Eq. 4

Eq 4 to Eq 3:
$2145 = 1.8y + y + 5y = 7.8y
y = $2145/7.8 = $275

Substitute y to Eq. 4 to solve for x:
x = 1.8y = 1.8($275) = $495
Substitute y to Eq 1 to solve for z:
z = 5y = 5($275) = $1375

Let's counter-check if we got the right answers
$2145 = x + y +z = $495 + $275 + $1375
$2145 = $2145 (Correct)

Thus, these are the deposits for each bank:

Bank of US: $495
7/4 Bank: $275
Catch Bank: $1375