Can anyone help me solve these systems of equations word problems?

Question

1) The dimes and quarters in Tim's piggy bank are worth $11.60. He has 32 more dimes than quarters. How many coins does he have?
2) If five times the smaller of two numbers is subtracted from twice the greater nomber, the result is 16. If the greaternumber is increased by three times the smaller number, the result is 63. What are the numbers?
Please help me solve these.

Answer 1

1) d : dimes; q : quarters; d = q + 32

(0.10d) + (0.25q) = 11.60
(0.10 (q + 32) + (0.25q) = 11.60
0.10q + 3.2 + 0.25q = 11.60
0.35q = 8.4
q = 24
d = q+32 = 56
total coints = 24 + 56 = 80 coints

2) x : greater number; y : smaller number
=> 2x - 5y = 16
=> x + 3y = 63 ==> x = 63 - 3y

2x - 5y = 16
2(63 - 3y) - 5y = 16
126 - 6y - 5y = 16
11y = 110
y = 10
x = 63 - 3y = 63 - 30 = 33

so, the greater number is 33 and the smaller number is 10.

Answer 2

1) Let x be the number of quarters. Then x+32 is the number of dimes. Since the total they add up to is 11.60, your equation is
value of quarters + value of dimes = 11.60
.25x + .10(x+32) = 11.60

solve for x to get number of quarters. Then x+32 is the number of times. Total coins is number of quarters + number of dimes.

2) let x be the smaller number, y the larger.

2y - 5x = 16
y + 3x = 63 so y = 63 - 3x then substitute for y in first equation

2(63 - 3x) - 5x = 16
126 - 6x - 5x = 16
-11x = 16 - 126

you can solve it from there for x - then put that number into one of the original equations so solve for y, the other number.

Answer 3

1)
suppose there are X dimes and Y quarters
total value of them is $11.6 = 1160 cents

==> 10X+25Y=1160
also 32 more dimes then quarters
==> Y=X-32

==>10X + 25 (X-32) = 1160
10X+25X - 800=1160
35X= 1160+800= 1960
X=56
==> Y= X-32 = 56-32 = 24

total coins = 56+24 = 80
2) let x be the smaller number and y be the greater number
2y-5x=16 and
y+3x=63 multiplying this equation by 2 ==> 2y+6x = 126

eq 2 - eq 3 gives
2y+6x=126
- (2y-5x=16)
------------------------
11x=110
x=10
solving gives y= (16+50)/2 = 33

x=10 y=33

Answer 4

x = dimes
y = quarters
.10x + .25 y = 11.60 | dimes + quarters = 11.60
x = y + 32 | 32 more dimes than quarters.
.10y + 3.20 + .25 y == .35y + 3.20 = 11.60 | Substitution
.35y = 8.40 | subtract 3.2 from both sides
If Y = 24, .35Y = 8.40
Y = 24, X = 56

Answer 5

a.) 24 quarters and 56 dimes

b.) the smaller number is 10 and the greater number is 33