Whakaoti i te {l}{4x+9y=-16}{10x+6y=26}

$$\left. \begin{array} { l } { 4 x + 9 y = - 16 } \\ { 10 x + 6 y = 26 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-4$$

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.

$$4x+9y=-16,10x+6y=26$$

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

$$4x+9y=-16$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$4x=-9y-16$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-9y-16\right)$$

Whakareatia \frac{1}{4} ki te -9y-16.

$$x=-\frac{9}{4}y-4$$

Whakakapia te -\frac{9y}{4}-4 mō te x ki tērā atu whārite, 10x+6y=26.

$$10\left(-\frac{9}{4}y-4\right)+6y=26$$

Whakareatia 10 ki te -\frac{9y}{4}-4.

$$-\frac{45}{2}y-40+6y=26$$

Tāpiri -\frac{45y}{2} ki te 6y.

$$-\frac{33}{2}y-40=26$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$-\frac{33}{2}y=66$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{33}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{9}{4}y-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{4}\left(-4\right)-4$$

Whakareatia -\frac{9}{4} ki te -4.

$$x=9-4$$

Tāpiri -4 ki te 9.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-4$$

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.

$$4x+9y=-16,10x+6y=26$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&9\\10&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\26\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&9\\10&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&9\\10&6\end{matrix}\right))\left(\begin{matrix}4&9\\10&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\10&6\end{matrix}\right))\left(\begin{matrix}-16\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\10&6\end{matrix}\right))\left(\begin{matrix}-16\\26\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\10&6\end{matrix}\right))\left(\begin{matrix}-16\\26\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{4\times 6-9\times 10}&-\frac{9}{4\times 6-9\times 10}\\-\frac{10}{4\times 6-9\times 10}&\frac{4}{4\times 6-9\times 10}\end{matrix}\right)\left(\begin{matrix}-16\\26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{11}&\frac{3}{22}\\\frac{5}{33}&-\frac{2}{33}\end{matrix}\right)\left(\begin{matrix}-16\\26\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{11}\left(-16\right)+\frac{3}{22}\times 26\\\frac{5}{33}\left(-16\right)-\frac{2}{33}\times 26\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-4$$

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.

$$4x+9y=-16,10x+6y=26$$

Kia ōrite ai a 4x me 10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$10\times 4x+10\times 9y=10\left(-16\right),4\times 10x+4\times 6y=4\times 26$$

Whakarūnātia.

$$40x+90y=-160,40x+24y=104$$

Me tango 40x+24y=104 mai i 40x+90y=-160 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$40x-40x+90y-24y=-160-104$$

Tāpiri 40x ki te -40x. Ka whakakore atu ngā kupu 40x me -40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$90y-24y=-160-104$$

Tāpiri 90y ki te -24y.

$$66y=-160-104$$

Tāpiri -160 ki te -104.

$$66y=-264$$

Whakawehea ngā taha e rua ki te 66.

$$y=-4$$

Whakaurua te -4 mō y ki 10x+6y=26. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$10x+6\left(-4\right)=26$$

Whakareatia 6 ki te -4.

$$10x-24=26$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$10x=50$$

Whakawehea ngā taha e rua ki te 10.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-4$$