Whakaoti i te {l}{4x+5y=8}{-x+2y=-15}

$$\left. \begin{array} { l } { 4 x + 5 y = 8 } \\ { - x + 2 y = - 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=-4$$

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.

$$4x+5y=8,-x+2y=-15$$

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

$$4x+5y=8$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$4x=-5y+8$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+8\right)$$

Whakareatia \frac{1}{4} ki te -5y+8.

$$x=-\frac{5}{4}y+2$$

Whakakapia te -\frac{5y}{4}+2 mō te x ki tērā atu whārite, -x+2y=-15.

$$-\left(-\frac{5}{4}y+2\right)+2y=-15$$

Whakareatia -1 ki te -\frac{5y}{4}+2.

$$\frac{5}{4}y-2+2y=-15$$

Tāpiri \frac{5y}{4} ki te 2y.

$$\frac{13}{4}y-2=-15$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$\frac{13}{4}y=-13$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{4}, 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{5}{4}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\left(-4\right)+2$$

Whakareatia -\frac{5}{4} ki te -4.

$$x=5+2$$

Tāpiri 2 ki te 5.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=-4$$

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.

$$4x+5y=8,-x+2y=-15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\-1&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\-1&2\end{matrix}\right))\left(\begin{matrix}4&5\\-1&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\-15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\-1&2\end{matrix}\right))\left(\begin{matrix}8\\-15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-5\left(-1\right)}&-\frac{5}{4\times 2-5\left(-1\right)}\\-\frac{-1}{4\times 2-5\left(-1\right)}&\frac{4}{4\times 2-5\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}8\\-15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&-\frac{5}{13}\\\frac{1}{13}&\frac{4}{13}\end{matrix}\right)\left(\begin{matrix}8\\-15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times 8-\frac{5}{13}\left(-15\right)\\\frac{1}{13}\times 8+\frac{4}{13}\left(-15\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=-4$$

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.

$$4x+5y=8,-x+2y=-15$$

Kia ōrite ai a 4x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-4x-5y=-8,4\left(-1\right)x+4\times 2y=4\left(-15\right)$$

Whakarūnātia.

$$-4x-5y=-8,-4x+8y=-60$$

Me tango -4x+8y=-60 mai i -4x-5y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-5y-8y=-8+60$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-8y=-8+60$$

Tāpiri -5y ki te -8y.

$$-13y=-8+60$$

Tāpiri -8 ki te 60.

$$-13y=52$$

Whakawehea ngā taha e rua ki te -13.

$$y=-4$$

Whakaurua te -4 mō y ki -x+2y=-15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+2\left(-4\right)=-15$$

Whakareatia 2 ki te -4.

$$-x-8=-15$$

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

$$-x=-7$$

Whakawehea ngā taha e rua ki te -1.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=-4$$