Whakaoti i te {r}{x+4y=25}{-4x+3y=52}

$$\left. \begin{array} { r } { x + 4 y = 25 } \\ { - 4 x + 3 y = 52 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=8$$

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.

$$x+4y=25,-4x+3y=52$$

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

$$x+4y=25$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$x=-4y+25$$

Whakakapia te -4y+25 mō te x ki tērā atu whārite, -4x+3y=52.

$$-4\left(-4y+25\right)+3y=52$$

Whakareatia -4 ki te -4y+25.

$$16y-100+3y=52$$

Tāpiri 16y ki te 3y.

$$19y-100=52$$

Me tāpiri 100 ki ngā taha e rua o te whārite.

$$19y=152$$

Whakawehea ngā taha e rua ki te 19.

$$y=8$$

Whakaurua te 8 mō y ki x=-4y+25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times 8+25$$

Whakareatia -4 ki te 8.

$$x=-32+25$$

Tāpiri 25 ki te -32.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=8$$

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.

$$x+4y=25,-4x+3y=52$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&4\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\52\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&4\\-4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&4\\-4&3\end{matrix}\right))\left(\begin{matrix}1&4\\-4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-4&3\end{matrix}\right))\left(\begin{matrix}25\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-4&3\end{matrix}\right))\left(\begin{matrix}25\\52\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&4\\-4&3\end{matrix}\right))\left(\begin{matrix}25\\52\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3-4\left(-4\right)}&-\frac{4}{3-4\left(-4\right)}\\-\frac{-4}{3-4\left(-4\right)}&\frac{1}{3-4\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}25\\52\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}&-\frac{4}{19}\\\frac{4}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}25\\52\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}\times 25-\frac{4}{19}\times 52\\\frac{4}{19}\times 25+\frac{1}{19}\times 52\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=8$$

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.

$$x+4y=25,-4x+3y=52$$

Kia ōrite ai a x 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 1.

$$-4x-4\times 4y=-4\times 25,-4x+3y=52$$

Whakarūnātia.

$$-4x-16y=-100,-4x+3y=52$$

Me tango -4x+3y=52 mai i -4x-16y=-100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x-16y-3y=-100-52$$

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.

$$-16y-3y=-100-52$$

Tāpiri -16y ki te -3y.

$$-19y=-100-52$$

Tāpiri -100 ki te -52.

$$-19y=-152$$

Whakawehea ngā taha e rua ki te -19.

$$y=8$$

Whakaurua te 8 mō y ki -4x+3y=52. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+3\times 8=52$$

Whakareatia 3 ki te 8.

$$-4x+24=52$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-4x=28$$

Whakawehea ngā taha e rua ki te -4.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=8$$