Whakaoti i te {l}{7x+8y=30}{8x-5y=6}

$$\left. \begin{array} { l } { 7 x + 8 y = 30 } \\ { 8 x - 5 y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$7x+8y=30,8x-5y=6$$

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

$$7x+8y=30$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$7x=-8y+30$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-8y+30\right)$$

Whakareatia \frac{1}{7} ki te -8y+30.

$$x=-\frac{8}{7}y+\frac{30}{7}$$

Whakakapia te \frac{-8y+30}{7} mō te x ki tērā atu whārite, 8x-5y=6.

$$8\left(-\frac{8}{7}y+\frac{30}{7}\right)-5y=6$$

Whakareatia 8 ki te \frac{-8y+30}{7}.

$$-\frac{64}{7}y+\frac{240}{7}-5y=6$$

Tāpiri -\frac{64y}{7} ki te -5y.

$$-\frac{99}{7}y+\frac{240}{7}=6$$

Me tango \frac{240}{7} mai i ngā taha e rua o te whārite.

$$-\frac{99}{7}y=-\frac{198}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{99}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{8}{7}y+\frac{30}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{8}{7}\times 2+\frac{30}{7}$$

Whakareatia -\frac{8}{7} ki te 2.

$$x=\frac{-16+30}{7}$$

Tāpiri \frac{30}{7} ki te -\frac{16}{7} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$7x+8y=30,8x-5y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&8\\8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&8\\8&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&8\\8&-5\end{matrix}\right))\left(\begin{matrix}7&8\\8&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\8&-5\end{matrix}\right))\left(\begin{matrix}30\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\8&-5\end{matrix}\right))\left(\begin{matrix}30\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\8&-5\end{matrix}\right))\left(\begin{matrix}30\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{7\left(-5\right)-8\times 8}&-\frac{8}{7\left(-5\right)-8\times 8}\\-\frac{8}{7\left(-5\right)-8\times 8}&\frac{7}{7\left(-5\right)-8\times 8}\end{matrix}\right)\left(\begin{matrix}30\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{99}&\frac{8}{99}\\\frac{8}{99}&-\frac{7}{99}\end{matrix}\right)\left(\begin{matrix}30\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{99}\times 30+\frac{8}{99}\times 6\\\frac{8}{99}\times 30-\frac{7}{99}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$7x+8y=30,8x-5y=6$$

Kia ōrite ai a 7x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$8\times 7x+8\times 8y=8\times 30,7\times 8x+7\left(-5\right)y=7\times 6$$

Whakarūnātia.

$$56x+64y=240,56x-35y=42$$

Me tango 56x-35y=42 mai i 56x+64y=240 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$56x-56x+64y+35y=240-42$$

Tāpiri 56x ki te -56x. Ka whakakore atu ngā kupu 56x me -56x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$64y+35y=240-42$$

Tāpiri 64y ki te 35y.

$$99y=240-42$$

Tāpiri 240 ki te -42.

$$99y=198$$

Whakawehea ngā taha e rua ki te 99.

$$y=2$$

Whakaurua te 2 mō y ki 8x-5y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-5\times 2=6$$

Whakareatia -5 ki te 2.

$$8x-10=6$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$8x=16$$

Whakawehea ngā taha e rua ki te 8.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=2$$