Whakaoti i te {l}{(A+B)3-B=6}{(2A+B)3^2-B=42}

$$\left. \begin{array} { l } { ( A + B ) 3 - B = 6 } \\ { ( 2 A + B ) 3 ^ { 2 } - B = 42 } \end{array} \right.$$

Whakaoti mō A, B

$$A=3$$

$$B = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te A+B ki te 3.

$$3A+3B-B=6$$

Pahekotia te 3B me -B, ka 2B.

$$3A+2B=6$$

Whakaarohia te whārite tuarua. Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$\left(2A+B\right)\times 9-B=42$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2A+B ki te 9.

$$18A+9B-B=42$$

Pahekotia te 9B me -B, ka 8B.

$$18A+8B=42$$

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$3A+2B=6,18A+8B=42$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te A mā te wehe i te A i te taha mauī o te tohu ōrite.

$$3A+2B=6$$

Me tango 2B mai i ngā taha e rua o te whārite.

$$3A=-2B+6$$

Whakawehea ngā taha e rua ki te 3.

$$A=\frac{1}{3}\left(-2B+6\right)$$

Whakareatia \frac{1}{3} ki te -2B+6.

$$A=-\frac{2}{3}B+2$$

Whakakapia te -\frac{2B}{3}+2 mō te A ki tērā atu whārite, 18A+8B=42.

$$18\left(-\frac{2}{3}B+2\right)+8B=42$$

Whakareatia 18 ki te -\frac{2B}{3}+2.

$$-12B+36+8B=42$$

Tāpiri -12B ki te 8B.

$$-4B+36=42$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$-4B=6$$

Whakawehea ngā taha e rua ki te -4.

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

Whakaurua te -\frac{3}{2} mō B ki A=-\frac{2}{3}B+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$A=-\frac{2}{3}\left(-\frac{3}{2}\right)+2$$

Whakareatia -\frac{2}{3} ki te -\frac{3}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$A=1+2$$

Tāpiri 2 ki te 1.

$$A=3$$

Kua oti te pūnaha te whakatau.

$$A=3,B=-\frac{3}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te A+B ki te 3.

$$3A+3B-B=6$$

Pahekotia te 3B me -B, ka 2B.

$$3A+2B=6$$

Whakaarohia te whārite tuarua. Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$\left(2A+B\right)\times 9-B=42$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2A+B ki te 9.

$$18A+9B-B=42$$

Pahekotia te 9B me -B, ka 8B.

$$18A+8B=42$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3A+2B=6,18A+8B=42$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&2\\18&8\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}6\\42\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\18&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\18&8\end{matrix}\right))\left(\begin{matrix}3&2\\18&8\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\18&8\end{matrix}\right))\left(\begin{matrix}6\\42\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\18&8\end{matrix}\right))\left(\begin{matrix}6\\42\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\18&8\end{matrix}\right))\left(\begin{matrix}6\\42\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3\times 8-2\times 18}&-\frac{2}{3\times 8-2\times 18}\\-\frac{18}{3\times 8-2\times 18}&\frac{3}{3\times 8-2\times 18}\end{matrix}\right)\left(\begin{matrix}6\\42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&\frac{1}{6}\\\frac{3}{2}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}6\\42\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 6+\frac{1}{6}\times 42\\\frac{3}{2}\times 6-\frac{1}{4}\times 42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}A\\B\end{matrix}\right)=\left(\begin{matrix}3\\-\frac{3}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa A me B.

$$A=3,B=-\frac{3}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te A+B ki te 3.

$$3A+3B-B=6$$

Pahekotia te 3B me -B, ka 2B.

$$3A+2B=6$$

Whakaarohia te whārite tuarua. Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$\left(2A+B\right)\times 9-B=42$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2A+B ki te 9.

$$18A+9B-B=42$$

Pahekotia te 9B me -B, ka 8B.

$$18A+8B=42$$

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$3A+2B=6,18A+8B=42$$

Kia ōrite ai a 3A me 18A, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 18 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$18\times 3A+18\times 2B=18\times 6,3\times 18A+3\times 8B=3\times 42$$

Whakarūnātia.

$$54A+36B=108,54A+24B=126$$

Me tango 54A+24B=126 mai i 54A+36B=108 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$54A-54A+36B-24B=108-126$$

Tāpiri 54A ki te -54A. Ka whakakore atu ngā kupu 54A me -54A, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36B-24B=108-126$$

Tāpiri 36B ki te -24B.

$$12B=108-126$$

Tāpiri 108 ki te -126.

$$12B=-18$$

Whakawehea ngā taha e rua ki te 12.

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

Whakaurua te -\frac{3}{2} mō B ki 18A+8B=42. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō A hāngai tonu.

$$18A+8\left(-\frac{3}{2}\right)=42$$

Whakareatia 8 ki te -\frac{3}{2}.

$$18A-12=42$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$18A=54$$

Whakawehea ngā taha e rua ki te 18.

$$A=3$$

Kua oti te pūnaha te whakatau.

$$A=3,B=-\frac{3}{2}$$