تمارين في الرياضيات

حل التمرين 1:

حساب A:

$$ A = (-\frac{6}{5} \times \frac{10}{3}) + (\frac{18}{7} \times \frac{14}{9}) $$

$$ A = (-\frac{60}{15}) + (\frac{252}{63}) $$

$$ A = -4 + 4 $$

$$ A = 0 $$

حساب B:

$$ B = (\frac{16}{55} \times \frac{37}{15}) - (\frac{37}{15} \times \frac{39}{55}) $$

$$ B = \frac{37}{15} \times (\frac{16}{55} - \frac{39}{55}) $$

$$ B = \frac{37}{15} \times (-\frac{23}{55}) $$

$$ B = -\frac{851}{825} $$

حساب C:

$$ C = (\frac{19}{31} \times \frac{31}{19}) + 31 $$

$$ C = 1 + 31 $$

$$ C = 32 $$

حساب E:

$$ E = \frac{\frac{5}{2} - \frac{3}{4}}{\frac{1}{5} + \frac{1}{3}} $$

$$ E = \frac{\frac{10}{4} - \frac{3}{4}}{\frac{3}{15} + \frac{5}{15}} $$

$$ E = \frac{\frac{7}{4}}{\frac{8}{15}} $$

$$ E = \frac{7}{4} \times \frac{15}{8} $$

$$ E = \frac{105}{32} $$

حساب F:

$$ F = \frac{\frac{3}{2} \times \frac{2}{3}}{5 - 2} $$

$$ F = \frac{1}{3} $$

حساب G:

$$ G = \frac{13 \times 12}{51 \times 5} $$

$$ G = \frac{156}{255} $$

$$ G = \frac{52}{85} $$ (بعد التبسيط بالقسمة على 3)

حساب H:

$$ H = \frac{1}{2} - \frac{\frac{3}{5} \times \frac{11}{-3}}{-4} $$

$$ H = \frac{1}{2} - \frac{-\frac{11}{5}}{-4} $$

$$ H = \frac{1}{2} - (-\frac{11}{5} \times -\frac{1}{4}) $$

$$ H = \frac{1}{2} - \frac{11}{20} $$

$$ H = \frac{10}{20} - \frac{11}{20} $$

$$ H = -\frac{1}{20} $$

حساب K:

$$ K = \frac{3}{3 - \frac{1}{3}} $$

$$ K = \frac{3}{\frac{9}{3} - \frac{1}{3}} $$

$$ K = \frac{3}{\frac{8}{3}} $$

$$ K = 3 \times \frac{3}{8} $$

$$ K = \frac{9}{8} $$

حل التمرين 2:

تبسيط العبارة E أولاً:

$$ E = \frac{\frac{1}{a} - \frac{1}{b}}{a \times b} $$

$$ E = \frac{\frac{b - a}{a \times b}}{a \times b} $$

$$ E = \frac{b - a}{(a \times b)^2} $$

أ) حساب E إذا علمت أن a = -3 و b = 3/2:

$$ b - a = \frac{3}{2} - (-3) = \frac{3}{2} + 3 = \frac{3}{2} + \frac{6}{2} = \frac{9}{2} $$

$$ a \times b = (-3) \times \frac{3}{2} = -\frac{9}{2} $$

$$ (a \times b)^2 = (-\frac{9}{2})^2 = \frac{81}{4} $$

$$ E = \frac{\frac{9}{2}}{\frac{81}{4}} $$

$$ E = \frac{9}{2} \times \frac{4}{81} $$

$$ E = \frac{36}{162} $$

$$ E = \frac{2}{9} $$ (بعد التبسيط بالقسمة على 18)

ب) حساب E إذا علمت أن b = -1/2 و a = -5/2:

$$ b - a = -\frac{1}{2} - (-\frac{5}{2}) = -\frac{1}{2} + \frac{5}{2} = \frac{4}{2} = 2 $$

$$ a \times b = (-\frac{5}{2}) \times (-\frac{1}{2}) = \frac{5}{4} $$

$$ (a \times b)^2 = (\frac{5}{4})^2 = \frac{25}{16} $$

$$ E = \frac{2}{\frac{25}{16}} $$

$$ E = 2 \times \frac{16}{25} $$

$$ E = \frac{32}{25} $$

حل التمرين 3:

حساب A:

$$ A = -\frac{5}{3} \times |-2 + \frac{2}{3}| + \frac{1}{2} $$

$$ A = -\frac{5}{3} \times |-\frac{6}{3} + \frac{2}{3}| + \frac{1}{2} $$

$$ A = -\frac{5}{3} \times |-\frac{4}{3}| + \frac{1}{2} $$

$$ A = -\frac{5}{3} \times \frac{4}{3} + \frac{1}{2} $$

$$ A = -\frac{20}{9} + \frac{1}{2} $$

$$ A = -\frac{40}{18} + \frac{9}{18} $$

$$ A = -\frac{31}{18} $$

حساب B:

$$ B = -\frac{13}{2} \times \frac{5}{2} + \frac{13}{2} \times (-\frac{4}{3}) $$

$$ B = \frac{13}{2} \times (-\frac{5}{2} - \frac{4}{3}) $$

$$ B = \frac{13}{2} \times (-\frac{15}{6} - \frac{8}{6}) $$

$$ B = \frac{13}{2} \times (-\frac{23}{6}) $$

$$ B = -\frac{299}{12} $$

حساب C:

$$ C = \frac{3}{5} - \frac{3}{-2} $$

$$ C = \frac{3}{5} + \frac{3}{2} $$

$$ C = \frac{6}{10} + \frac{15}{10} $$

$$ C = \frac{21}{10} $$

حساب D:

$$ D = \frac{1}{- \frac{7}{3}} \times (-2) + 2 $$

$$ D = -\frac{3}{7} \times (-2) + 2 $$

$$ D = \frac{6}{7} + 2 $$

$$ D = \frac{6}{7} + \frac{14}{7} $$

$$ D = \frac{20}{7} $$

حل التمرين 4:

الحالة الأولى:

$$ -\frac{2}{3} |2x - 5| = -\frac{3}{2} $$

$$ |2x - 5| = (-\frac{3}{2}) \div (-\frac{2}{3}) $$

$$ |2x - 5| = \frac{9}{4} $$

إما: $$ 2x - 5 = \frac{9}{4} $$

$$ 2x = \frac{9}{4} + 5 $$

$$ 2x = \frac{9}{4} + \frac{20}{4} $$

$$ 2x = \frac{29}{4} $$

$$ x = \frac{29}{8} $$

أو: $$ 2x - 5 = -\frac{9}{4} $$

$$ 2x = -\frac{9}{4} + 5 $$

$$ 2x = -\frac{9}{4} + \frac{20}{4} $$

$$ 2x = \frac{11}{4} $$

$$ x = \frac{11}{8} $$

الحالة الثانية:

$$ -\frac{3}{4} (2x - \frac{1}{3}) = \frac{5}{2} $$

$$ 2x - \frac{1}{3} = \frac{5}{2} \div (-\frac{3}{4}) $$

$$ 2x - \frac{1}{3} = -\frac{20}{6} $$

$$ 2x - \frac{1}{3} = -\frac{10}{3} $$

$$ 2x = -\frac{10}{3} + \frac{1}{3} $$

$$ 2x = -\frac{9}{3} $$

$$ 2x = -3 $$

$$ x = -\frac{3}{2} $$

حل التمرين 5:

حساب E:

$$ E = -\frac{3}{5}x - (-\frac{3}{5}y) + \frac{2}{7} $$

$$ E = -\frac{3}{5}x + \frac{3}{5}y + \frac{2}{7} $$

$$ E = -\frac{3}{5}(x - y) + \frac{2}{7} $$

بما أن $$ x - y = -\frac{3}{7} $$:

$$ E = -\frac{3}{5} \times (-\frac{3}{7}) + \frac{2}{7} $$

$$ E = \frac{9}{35} + \frac{2}{7} $$

$$ E = \frac{9}{35} + \frac{10}{35} $$

$$ E = \frac{19}{35} $$

حساب F:

$$ F = \frac{x}{5} + \frac{y}{-2} $$

$$ F = \frac{x}{5} - \frac{y}{2} $$

$$ F = \frac{2x - 5y}{10} $$

من المعطى $$ 5y - 2x = -15 $$، يمكننا كتابة $$ 2x - 5y = 15 $$:

$$ F = \frac{15}{10} $$

$$ F = \frac{3}{2} $$

حل التمرين 6:

لدينا: $$ M = (a - \frac{6}{5})(2 - \frac{5}{3}b) - \frac{3}{4} a \times (-\frac{8}{3})b $$

نقوم بتبسيط الجزء الثاني أولاً:

$$ -\frac{3}{4} a \times (-\frac{8}{3})b = \frac{24}{12}ab = 2ab $$

إذن، العبارة M تصبح: $$ M = (a - \frac{6}{5})(2 - \frac{5}{3}b) + 2ab $$

من العلاقة: $$ \frac{a}{-5} = \frac{2}{b} $$

بضرب الطرفين في الوسطين: $$ ab = -10 $$

نعوض قيمة ab في M:

$$ M = (a - \frac{6}{5})(2 - \frac{5}{3}b) + 2(-10) $$

$$ M = (a - \frac{6}{5})(2 - \frac{5}{3}b) - 20 $$

نقوم الآن بنشر الجزء الأول:

$$ M = 2a - \frac{5}{3}ab - \frac{12}{5} + 2b - 20 $$

نعوض $$ ab = -10 $$:

$$ M = 2a + \frac{50}{3} - \frac{12}{5} + 2b - 20 $$

نجمع الحدود الثابتة:

$$ M = 2(a+b) + \frac{250}{15} - \frac{36}{15} - \frac{300}{15} $$

$$ M = 2(a+b) - \frac{86}{15} $$

بما أن $$ a+b = -\frac{3}{2} $$:

$$ M = 2(-\frac{3}{2}) - \frac{86}{15} $$

$$ M = -3 - \frac{86}{15} $$

$$ M = -\frac{45}{15} - \frac{86}{15} $$

$$ M = -\frac{131}{15} $$