Question

43 coins, no nickels, $4.51 equal amount amount of pennies and dimes, how many of each coin

Answer 1

Set up
Let the dimes = d
Let the pennies = p
Let the quarters = q

Equations
You cannot mean that the pennies and dimes have equal numbers. That would mean that each had 21.5 members. Now could you mean that the dime and penny amount could be the same with 43 coins that total 4.00. Four dollars means that you need 40 dimes alone. It must mean that you are including quarters. 

p + d + p = 43          (1)
p = d                        (2)
p +10d +25q = 451  (3)

Note how this last equation = was derived. You have to multiply the dimes by 10 and he quarters by 100 and the total by 100 to get the numbers all in pennies.

Put the results of 2 into 1.
2p  + q = 43                   (4)  
You need to modify equation 3 as well.
p + 10p + 25q = 451
11p + 25q  = 451           (5)

Solve the new equations
2p +  q = 43               (4)
11p + 25q = 451        (5)

Multiply 4 by 25
25(2p- + q = 43)
50p + 25q = 1075     (6)  Subtract (5) from (6)
11p + 25q = 451
39p = 624                        Divide by 39
p = 624 / 39
p = 16

Since the pennies and dimes are equal there are 16 dimes
p + d + q = 43
16 + 16 + q = 43
32 + q = 43
q = 11

Check
16 + 10*16 + 11*25 = ?
16 + 160 + 275  = ?
451 = ?

Nice problem. Thanks for posting.