Whakaoti i te {l}{7x-8y=9}{4x-13y=-10}

$$\left\{ \begin{array} { l } { 7 x - 8 y = 9 } \\ { 4 x - 13 y = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{197}{59} = 3\frac{20}{59} \approx 3.338983051$$

$$y = \frac{106}{59} = 1\frac{47}{59} \approx 1.796610169$$

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.

$$7x-8y=9,4x-13y=-10$$

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

$$7x-8y=9$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$7x=8y+9$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(8y+9\right)$$

Whakareatia \frac{1}{7} ki te 8y+9.

$$x=\frac{8}{7}y+\frac{9}{7}$$

Whakakapia te \frac{8y+9}{7} mō te x ki tērā atu whārite, 4x-13y=-10.

$$4\left(\frac{8}{7}y+\frac{9}{7}\right)-13y=-10$$

Whakareatia 4 ki te \frac{8y+9}{7}.

$$\frac{32}{7}y+\frac{36}{7}-13y=-10$$

Tāpiri \frac{32y}{7} ki te -13y.

$$-\frac{59}{7}y+\frac{36}{7}=-10$$

Me tango \frac{36}{7} mai i ngā taha e rua o te whārite.

$$-\frac{59}{7}y=-\frac{106}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{59}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{106}{59}$$

Whakaurua te \frac{106}{59} mō y ki x=\frac{8}{7}y+\frac{9}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{7}\times \frac{106}{59}+\frac{9}{7}$$

Whakareatia \frac{8}{7} ki te \frac{106}{59} 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.

$$x=\frac{848}{413}+\frac{9}{7}$$

Tāpiri \frac{9}{7} ki te \frac{848}{413} 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.

$$x=\frac{197}{59}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{197}{59},y=\frac{106}{59}$$

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.

$$7x-8y=9,4x-13y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-8\\4&-13\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right))\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-8\\4&-13\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{7\left(-13\right)-\left(-8\times 4\right)}&-\frac{-8}{7\left(-13\right)-\left(-8\times 4\right)}\\-\frac{4}{7\left(-13\right)-\left(-8\times 4\right)}&\frac{7}{7\left(-13\right)-\left(-8\times 4\right)}\end{matrix}\right)\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{59}&-\frac{8}{59}\\\frac{4}{59}&-\frac{7}{59}\end{matrix}\right)\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{59}\times 9-\frac{8}{59}\left(-10\right)\\\frac{4}{59}\times 9-\frac{7}{59}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{197}{59}\\\frac{106}{59}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{197}{59},y=\frac{106}{59}$$

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.

$$7x-8y=9,4x-13y=-10$$

Kia ōrite ai a 7x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$4\times 7x+4\left(-8\right)y=4\times 9,7\times 4x+7\left(-13\right)y=7\left(-10\right)$$

Whakarūnātia.

$$28x-32y=36,28x-91y=-70$$

Me tango 28x-91y=-70 mai i 28x-32y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x-32y+91y=36+70$$

Tāpiri 28x ki te -28x. Ka whakakore atu ngā kupu 28x me -28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-32y+91y=36+70$$

Tāpiri -32y ki te 91y.

$$59y=36+70$$

Tāpiri 36 ki te 70.

$$59y=106$$

Whakawehea ngā taha e rua ki te 59.

$$y=\frac{106}{59}$$

Whakaurua te \frac{106}{59} mō y ki 4x-13y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-13\times \frac{106}{59}=-10$$

Whakareatia -13 ki te \frac{106}{59}.

$$4x-\frac{1378}{59}=-10$$

Me tāpiri \frac{1378}{59} ki ngā taha e rua o te whārite.

$$4x=\frac{788}{59}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{197}{59}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{197}{59},y=\frac{106}{59}$$