Question

15. Determinați câte trei perechi de numere naturale a și b pentru care au loc relațiile: a) 11/15 - 3/20 = a/3 - b/12 b). 9/14 - 5/21 = a/2 - b/21 c) 11/24 - a/20 = b/8 - 4/15

Answer 1

aducem la același numitor:

a)

$\dfrac{^{4)} 11}{15} - \dfrac{^{3)} 3}{20} = \dfrac{^{20)} a}{3} - \dfrac{^{5)} b}{12}$

$44 - 9 = 20a -5b$

$35 = 20a - 5b \ \ \Big|:5$

$4a - b = 7 \iff \bf 4a = b + 7$

$a = 2 \Leftrightarrow 8 = b + 7 \implies b = 1$

$a = 3 \Leftrightarrow 12 = b + 7 \implies b = 5$

$a = 4 \Leftrightarrow 16 = b + 7 \implies b = 9$

perechile sunt: (2;1), (3;5), (4;9)

b)

$\dfrac{^{3)} 9}{14} - \dfrac{^{2)} 5}{21} = \dfrac{^{21)} a}{2} - \dfrac{^{2)} b}{21}$

$27 - 10 = 21a - 2b$

$\bf 21a = 2b + 17$

$a = 1 \Leftrightarrow 21 = 2b + 17 \Leftrightarrow 2b = 4 \implies b = 2$

$a = 3 \Leftrightarrow 63 = 2b + 17 \Leftrightarrow 2b = 46 \implies b = 23$

$a = 5 \Leftrightarrow 105 = 2b + 17 \Leftrightarrow 2b = 88 \implies b = 44$

perechile sunt: (1;2), (3;23), (5;44)

c)

$\dfrac{^{5)} 11}{24} - \dfrac{^{6)} a}{20} = \dfrac{^{15)} b}{8} - \dfrac{^{8)} 4}{15}$

$55 - 6a = 15b - 32$

$\bf 6a = 87 - 15b$

$b = 1 \Leftrightarrow 6a = 87 - 15 \Leftrightarrow 6a = 72 \implies a = 12$

$b = 3 \Leftrightarrow 6a = 87 - 45 \Leftrightarrow 6a = 42 \implies a = 7$

$b = 5 \Leftrightarrow 6a = 87 - 75 \Leftrightarrow 6a = 12 \implies a = 2$

perechile sunt: (2;5), (7;3), (12;1)