1. {a,b,c} d.p. {2,3,4}
[tex] \frac{a}{2} = \frac{b}{3} = \frac{c}{3} = k [/tex]
k-coeficient de proportionalitate
[tex] \frac{a}{2} = k = > a = 2k[/tex]
[tex] \frac{b}{3} = k = > b = 3k[/tex]
[tex] \frac{c}{4} = k = > c = 4k[/tex]
a+b+c= 180
2k+3k+4k=180
9k=180
k=20
a=2k=>a=2×20=>a=40
b=3k=>b=3×20=>b=60
c=4k=>c=4×20=>c=80
2. {a,b} i.p. {3,7}
3a=7b=k
[tex]3a = k = > a = \frac{k}{3} [/tex]
[tex]7b = k = > b = \frac{k}{7} [/tex]
a+b=100
[tex] \frac{k}{3} + \frac{k}{7} = 100 \\ aduci \: la \: acelasi \: numitor - > 21 \\ \frac{7k}{21} + \frac{3k}{21} = 100 \\ \frac{10k}{21} = 100 \\ 10k = 21 \times 100 \\ 10k = 2100 \\ k = 2100 \div 10 \\ k = 210[/tex]
[tex]a = \frac{k}{3} = > a = \frac{210}{3} = 70[/tex]
[tex]b = \frac{k}{7} = > b = \frac{210}{7} = 30 [/tex]
3. {x,y,z} i.p.{1,5; 2,4; 3}
[tex]1.5 = \frac{15}{10} = \frac{3}{2} \\ 2.4 = \frac{24}{10} = \frac{12}{5} [/tex]
[tex] \frac{3}{2} \times x = \frac{12}{5} \times y = 3z = k \\ \frac{3x}{2} = \frac{12y}{5} = 3z = k[/tex]
[tex] \frac{3x}{2} = k = > 3x = 2k = > x = \frac{2k}{3} \\ \frac{12y}{5} = k = > 12y = 5k = > y = \frac{5k}{12} \\ 3z = k = > z = \frac{k}{3} [/tex]
4x-6y+z=6
[tex]4 \times \frac{2k}{3} - 6 \times \frac{5k}{12} + \frac{k}{3} = 6 \\ \frac{8k}{3} - \frac{30k}{12} + \frac{k}{3} = 6 \\ \frac{8k}{3} - \frac{5k}{2} + \frac{k}{3} = 6 \\ \frac{9k}{3} - \frac{5k}{2} = 6 \\ aduci \: la \: acelasi \: numtor \\ \frac{18k}{6} - \frac{15k}{6} = 6 \\ \frac{3k}{6} = 6 = > 3k = 6 \times 6 = > \\ 3k = 36 = > k = 12[/tex]
[tex]x = \frac{2k}{3} = \frac{2 \times 12}{3} = \frac{24}{3} = 8[/tex]
[tex]y = \frac{5k}{12} = \frac{5 \times 12}{12} = 5[/tex]
[tex]z = \frac{k}{3} = \frac{12}{3} = 4[/tex]
