There are 31 ways:1p × 211p × 19 + 2p × 11p × 17 + 2p × 21p × 16 + 5p × 11p × 15 + 2p × 31p × 14 + 2p × 1 + 5p × 11p × 13 + 2p × 41p × 12 + 2p × 2 + 5p × 11p × 11 + 2p × 51p × 11 + 5p × 21p × 10 + 2p × 3 + 5p × 11p × 9 + 2p × 61p × 9 + 2p × 1 + 5p × 21p × 8 + 2p × 4 + 5p1p × 7 + 2p × 71p × 7 + 2p × 2 + 5p × 21p × 6 + 2p × 5 + 5p1p × 6 + 5p × 31p × 5 + 2p × 81p × 5 + 2p × 3 + 5p × 21p × 4 + 2p × 6 + 5p × 11p × 4 + 2p × 1 + 5p × 31p × 3 + 2p × 91p × 3 + 2p × 4 + 5p × 21p × 2 + 2p × 7 + 5p × 11p × 2 + 2p × 2 + 5p × 31p × 1 + 2p × 101p × 1 + 2p × 5 + 5p × 21p × 1 + 5p × 42p × 8 + 5p × 12p × 3 + 5p × 3
6p and 5p are the coins
Reduce the second part by taking 8 out. (11p + 8)/6 = p - 2 Multiply both sides by 6. 11p + 8 = 6p - 12 Subtract 6p from both sides. 5p + 8 = -12 Subtract 8 from both sides. 5p = -20 Divide both sides by 5. p = -4 Check it by substituting -4 for p in the original equation. (-44 + 8)/6 = (-32 - 16)/8 -36/6 = -48/8 -6 = -6 It checks.
7p + 2 = 5p + 8 7p - 5p = 8 - 2 2p = 6 p = 3
5p + 4y
6p + 11q + 4
7p + 2q = 46 . . . . (A) 5p + 3q = 36 . . . . (B) 3*(A): 21p + 6q = 138 2*(B): 10p + 6q = 72 Subtracting gives 11p = 66 so that p = 6 Substitute for p in (A): 7*6 + 2q = 46 or 42 + 2q = 46 which gives 2q = 4 so that q = 2 Solution: (p, q) = (6,2)
There are 31 ways:1p × 211p × 19 + 2p × 11p × 17 + 2p × 21p × 16 + 5p × 11p × 15 + 2p × 31p × 14 + 2p × 1 + 5p × 11p × 13 + 2p × 41p × 12 + 2p × 2 + 5p × 11p × 11 + 2p × 51p × 11 + 5p × 21p × 10 + 2p × 3 + 5p × 11p × 9 + 2p × 61p × 9 + 2p × 1 + 5p × 21p × 8 + 2p × 4 + 5p1p × 7 + 2p × 71p × 7 + 2p × 2 + 5p × 21p × 6 + 2p × 5 + 5p1p × 6 + 5p × 31p × 5 + 2p × 81p × 5 + 2p × 3 + 5p × 21p × 4 + 2p × 6 + 5p × 11p × 4 + 2p × 1 + 5p × 31p × 3 + 2p × 91p × 3 + 2p × 4 + 5p × 21p × 2 + 2p × 7 + 5p × 11p × 2 + 2p × 2 + 5p × 31p × 1 + 2p × 101p × 1 + 2p × 5 + 5p × 21p × 1 + 5p × 42p × 8 + 5p × 12p × 3 + 5p × 3
6p and 5p are the coins
2p + 3q = 13, 5p - 4q = -2 Multiply the first equation by 4 and the second by 3 and add them, which gets rid of the q: 8p + 15p = 52 - 6, and 23p = 46, so p=2. Plug that into the first equation to find q: 4 + 3q = 13, so q=3. Test your answers in the second equation to be sure: 5(2) - 4(3) = 10-12 = -2. It checks. So p=2, q=3.
Reduce the second part by taking 8 out. (11p + 8)/6 = p - 2 Multiply both sides by 6. 11p + 8 = 6p - 12 Subtract 6p from both sides. 5p + 8 = -12 Subtract 8 from both sides. 5p = -20 Divide both sides by 5. p = -4 Check it by substituting -4 for p in the original equation. (-44 + 8)/6 = (-32 - 16)/8 -36/6 = -48/8 -6 = -6 It checks.
7p + 2 = 5p + 8 7p - 5p = 8 - 2 2p = 6 p = 3
5p + 4y
8 + 5p + 7q + 9 + 3p Reordering: 8 + 9 + 5p + 3p + 7q Combine like terms: 17 + 8p + 7q
5P + 111 + 136 + 20 = 4c + 4 5P + 267 = 4c + 4 -4 -4 5P + 263 = 4c You cannot cimplify it anymore, unless by 'c' you meant 'P'. 5P + 263 = 4P -5P -5P 263(-1) = -P(-1) -263 = P
10p + 5p - p = 15p - p = 14p
5p + 15p + 7p = 27p 5p - 15p + 7p = -3p 5p x 15p + 7p = 75p2 + 7p 5p/15p + 7p = 7p + 1/3