Whakaoti i te {r}{5x-3y=17}{-2x+5y=-22}

$$\left. \begin{array} { r } { 5 x - 3 y = 17 } \\ { - 2 x + 5 y = - 22 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-4$$

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.

$$5x-3y=17,-2x+5y=-22$$

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

$$5x-3y=17$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$5x=3y+17$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(3y+17\right)$$

Whakareatia \frac{1}{5} ki te 3y+17.

$$x=\frac{3}{5}y+\frac{17}{5}$$

Whakakapia te \frac{3y+17}{5} mō te x ki tērā atu whārite, -2x+5y=-22.

$$-2\left(\frac{3}{5}y+\frac{17}{5}\right)+5y=-22$$

Whakareatia -2 ki te \frac{3y+17}{5}.

$$-\frac{6}{5}y-\frac{34}{5}+5y=-22$$

Tāpiri -\frac{6y}{5} ki te 5y.

$$\frac{19}{5}y-\frac{34}{5}=-22$$

Me tāpiri \frac{34}{5} ki ngā taha e rua o te whārite.

$$\frac{19}{5}y=-\frac{76}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{5}, 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{3}{5}y+\frac{17}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{5}\left(-4\right)+\frac{17}{5}$$

Whakareatia \frac{3}{5} ki te -4.

$$x=\frac{-12+17}{5}$$

Tāpiri \frac{17}{5} ki te -\frac{12}{5} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-4$$

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.

$$5x-3y=17,-2x+5y=-22$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\-22\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-3\\-2&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right))\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right))\left(\begin{matrix}17\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right))\left(\begin{matrix}17\\-22\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-3\\-2&5\end{matrix}\right))\left(\begin{matrix}17\\-22\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{5\times 5-\left(-3\left(-2\right)\right)}&-\frac{-3}{5\times 5-\left(-3\left(-2\right)\right)}\\-\frac{-2}{5\times 5-\left(-3\left(-2\right)\right)}&\frac{5}{5\times 5-\left(-3\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}17\\-22\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}&\frac{3}{19}\\\frac{2}{19}&\frac{5}{19}\end{matrix}\right)\left(\begin{matrix}17\\-22\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}\times 17+\frac{3}{19}\left(-22\right)\\\frac{2}{19}\times 17+\frac{5}{19}\left(-22\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-4$$

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.

$$5x-3y=17,-2x+5y=-22$$

Kia ōrite ai a 5x me -2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-2\times 5x-2\left(-3\right)y=-2\times 17,5\left(-2\right)x+5\times 5y=5\left(-22\right)$$

Whakarūnātia.

$$-10x+6y=-34,-10x+25y=-110$$

Me tango -10x+25y=-110 mai i -10x+6y=-34 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x+6y-25y=-34+110$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-25y=-34+110$$

Tāpiri 6y ki te -25y.

$$-19y=-34+110$$

Tāpiri -34 ki te 110.

$$-19y=76$$

Whakawehea ngā taha e rua ki te -19.

$$y=-4$$

Whakaurua te -4 mō y ki -2x+5y=-22. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+5\left(-4\right)=-22$$

Whakareatia 5 ki te -4.

$$-2x-20=-22$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$-2x=-2$$

Whakawehea ngā taha e rua ki te -2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-4$$