Whakaoti i te {l}{5x-y=110}{9y-x=110}

$$\left\{ \begin{array} { l } { 5 x - y = 110 } \\ { 9 y - x = 110 } \end{array} \right.$$

Whakaoti mō x, y

$$x=25$$

$$y=15$$

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.

$$5x-y=110,-x+9y=110$$

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

$$5x-y=110$$

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

$$5x=y+110$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+110\right)$$

Whakareatia \frac{1}{5} ki te y+110.

$$x=\frac{1}{5}y+22$$

Whakakapia te \frac{y}{5}+22 mō te x ki tērā atu whārite, -x+9y=110.

$$-\left(\frac{1}{5}y+22\right)+9y=110$$

Whakareatia -1 ki te \frac{y}{5}+22.

$$-\frac{1}{5}y-22+9y=110$$

Tāpiri -\frac{y}{5} ki te 9y.

$$\frac{44}{5}y-22=110$$

Me tāpiri 22 ki ngā taha e rua o te whārite.

$$\frac{44}{5}y=132$$

Whakawehea ngā taha e rua o te whārite ki te \frac{44}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=15$$

Whakaurua te 15 mō y ki x=\frac{1}{5}y+22. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times 15+22$$

Whakareatia \frac{1}{5} ki te 15.

$$x=3+22$$

Tāpiri 22 ki te 3.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=15$$

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.

$$5x-y=110,-x+9y=110$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}110\\110\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\-1&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}110\\110\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}110\\110\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\-1&9\end{matrix}\right))\left(\begin{matrix}110\\110\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{5\times 9-\left(-\left(-1\right)\right)}&-\frac{-1}{5\times 9-\left(-\left(-1\right)\right)}\\-\frac{-1}{5\times 9-\left(-\left(-1\right)\right)}&\frac{5}{5\times 9-\left(-\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}110\\110\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{44}&\frac{1}{44}\\\frac{1}{44}&\frac{5}{44}\end{matrix}\right)\left(\begin{matrix}110\\110\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{44}\times 110+\frac{1}{44}\times 110\\\frac{1}{44}\times 110+\frac{5}{44}\times 110\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=25,y=15$$

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.

$$5x-y=110,-x+9y=110$$

Kia ōrite ai a 5x 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 5.

$$-5x-\left(-y\right)=-110,5\left(-1\right)x+5\times 9y=5\times 110$$

Whakarūnātia.

$$-5x+y=-110,-5x+45y=550$$

Me tango -5x+45y=550 mai i -5x+y=-110 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x+y-45y=-110-550$$

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-45y=-110-550$$

Tāpiri y ki te -45y.

$$-44y=-110-550$$

Tāpiri -110 ki te -550.

$$-44y=-660$$

Whakawehea ngā taha e rua ki te -44.

$$y=15$$

Whakaurua te 15 mō y ki -x+9y=110. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+9\times 15=110$$

Whakareatia 9 ki te 15.

$$-x+135=110$$

Me tango 135 mai i ngā taha e rua o te whārite.

$$-x=-25$$

Whakawehea ngā taha e rua ki te -1.

$$x=25$$

Kua oti te pūnaha te whakatau.

$$x=25,y=15$$