Canje-canje na ainihi na maganganu

A cikin wannan ɗaba'ar, za mu yi la'akari da manyan nau'ikan sauye-sauye iri ɗaya na maganganun algebra, tare da su tare da dabaru da misalai don nuna aikace-aikacen su a aikace. Manufar irin waɗannan sauye-sauye shine maye gurbin ainihin magana tare da daidai daidai.

Sake tsara sharuddan da dalilai

A kowane jimla, zaku iya sake tsara sharuɗɗan.

a + b = b + a

A kowane samfurin, zaku iya sake tsara abubuwan.

a ⋅ b = b ⋅ a

misalai:

• 1 + 2 = 2 + 1
• 128 ⋅ 32 = 32 ⋅ 128

Sharuɗɗan ƙungiyoyi (masu yawa)

Idan akwai fiye da sharuɗɗan 2 a cikin jimillar, ana iya haɗa su ta bakake. Idan an buƙata, zaku iya fara musanya su.

a + b + c + d = (a + c) + (b + d)

A cikin samfurin, zaku iya kuma haɗa abubuwan.

a ⋅ b ⋅ c ⋅ d = (a ⋅ d) ⋅ (b ⋅ c)

misalai:

• 15 + 6 + 5 + 4 = (15 + 5) + (6 + 4)
• 6 ⋅ 8 ⋅ 11 ⋅ 4 = (6⋅ 4⋅ 8) ⋅ 11

Ƙara, ragi, ninka ko rarraba ta lamba ɗaya

Idan an ƙara ko rage lamba ɗaya zuwa sassan biyu na ainihi, to ya kasance gaskiya.

If a + b = c + dsa'an nan (a + b) ± e = (c + d) ± e.

Har ila yau, ba za a keta daidaito ba idan an ninka ko raba sassan biyu da lamba ɗaya.

If a + b = c + dsa'an nan (a + b) ⋅/: e = (c + d) ⋅/: e.

misalai:

• 35 + 10 = 9 + 16 + 20 ⇒ (35 + 10) + 4 = (9 + 16 + 20) + 4
• 42 + 14 = 7 ⋅ 8 ⇒ (42 + 14) ⋅ 12 = (7 ⋅ 8) ⋅ 12

Maye gurbin Bambanci tare da Jimi (sau da yawa samfur)

Ana iya wakilta kowane bambanci azaman jimlar sharuɗɗa.

a - b = a + (-b)

Za'a iya amfani da dabara iri ɗaya ga rarrabuwa, watau maye gurbin akai-akai da samfur.

a: b = a ⋅ b^-1

misalai:

• 76-15-29 = 76 + (-15) + (-29)
• 42: 3 = 42 ⋅ 3^-1

Yin ayyukan lissafi

Kuna iya sauƙaƙe kalmar lissafi (wani lokaci mai mahimmanci) ta hanyar aiwatar da ayyukan lissafin (ƙari, ragi, ninkawa da rarraba), la'akari da abin da aka yarda da su gaba ɗaya. odar kisa:

• da farko muna tadawa zuwa iko, cire tushen, lissafin logarithms, trigonometric da sauran ayyuka;
• sa'an nan kuma mu yi ayyuka a cikin brackets;
• ƙarshe - daga hagu zuwa dama, yi sauran ayyukan. Yawan yawa da rarrabuwa suna fifiko akan ƙari da ragi. Wannan kuma ya shafi maganganu a cikin baka.

misalai:

• 14 + 6 ⋅ (35 - 16 ⋅ 2) + 11 ⋅ 3 = 14 + 18 + 33 = 65
• 20: 4 + 2 ⋅ (25 ⋅ 3 - 15) - 9 + 2 ⋅ 8 = 5 + 120 - 9 + 16 = 132

Fadada maƙarƙashiya

Za a iya cire mahaifa a cikin furcin lissafi. Ana yin wannan aikin bisa ga wasu - dangane da waɗanne alamomi ("plus", "raguwa", "yawan" ko "raba") suna gaban ko bayan maƙallan.

misalai:

• 117 + (90 - 74 - 38) = 117 + 90 - 74 - 38
• 1040 - (-218 - 409 + 192) = 1040 + 218 + 409 - 192
• 22⋅ (8+14) = 22 ⋅ 8 + 22 ⋅ 14
• 18: (4-6) = 18:4-18:6

Bakin Haɓaka Babban Factor

Idan duk sharuɗɗan da ke cikin furcin suna da ma'ana gama gari, ana iya fitar da shi daga maƙasudin, wanda sharuɗɗan da aka raba ta wannan factor za su kasance. Wannan dabara kuma ta shafi masu canji na zahiri.

misalai:

• 3 ⋅ 5 + 5 ⋅ 6 = 5⋅ (3+6)
• 28 + 56 - 77 = 7 ⋅ (4 + 8-11)
• 31x + 50x = x ⋅ (31 + 50)

Aikace-aikace na gajerun hanyoyin ninkawa

Hakanan zaka iya amfani da su don yin sauye-sauye iri ɗaya na maganganun algebra.

misalai:

• (31 + 4)² = 31² + 2 ⋅ 31 ⋅ 4 + 4² = 1225
• 26² - 7² = (26 - 7) ⋅ (26 + 7) = 627