Whakaoti i te {l}{2x+3y=53}{3x-y=19}

$$\left. \begin{array} { l } { 2 x + 3 y = 53 } \\ { 3 x - y = 19 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=11$$

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.

$$2x+3y=53,3x-y=19$$

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

$$2x+3y=53$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+53$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+53\right)$$

Whakareatia \frac{1}{2} ki te -3y+53.

$$x=-\frac{3}{2}y+\frac{53}{2}$$

Whakakapia te \frac{-3y+53}{2} mō te x ki tērā atu whārite, 3x-y=19.

$$3\left(-\frac{3}{2}y+\frac{53}{2}\right)-y=19$$

Whakareatia 3 ki te \frac{-3y+53}{2}.

$$-\frac{9}{2}y+\frac{159}{2}-y=19$$

Tāpiri -\frac{9y}{2} ki te -y.

$$-\frac{11}{2}y+\frac{159}{2}=19$$

Me tango \frac{159}{2} mai i ngā taha e rua o te whārite.

$$-\frac{11}{2}y=-\frac{121}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=11$$

Whakaurua te 11 mō y ki x=-\frac{3}{2}y+\frac{53}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 11+\frac{53}{2}$$

Whakareatia -\frac{3}{2} ki te 11.

$$x=\frac{-33+53}{2}$$

Tāpiri \frac{53}{2} ki te -\frac{33}{2} 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.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=11$$

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.

$$2x+3y=53,3x-y=19$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}53\\19\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\3&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\3&-1\end{matrix}\right))\left(\begin{matrix}2&3\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-1\end{matrix}\right))\left(\begin{matrix}53\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-1\end{matrix}\right))\left(\begin{matrix}53\\19\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&-1\end{matrix}\right))\left(\begin{matrix}53\\19\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2\left(-1\right)-3\times 3}&-\frac{3}{2\left(-1\right)-3\times 3}\\-\frac{3}{2\left(-1\right)-3\times 3}&\frac{2}{2\left(-1\right)-3\times 3}\end{matrix}\right)\left(\begin{matrix}53\\19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{3}{11}\\\frac{3}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}53\\19\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 53+\frac{3}{11}\times 19\\\frac{3}{11}\times 53-\frac{2}{11}\times 19\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\11\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=11$$

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.

$$2x+3y=53,3x-y=19$$

Kia ōrite ai a 2x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$3\times 2x+3\times 3y=3\times 53,2\times 3x+2\left(-1\right)y=2\times 19$$

Whakarūnātia.

$$6x+9y=159,6x-2y=38$$

Me tango 6x-2y=38 mai i 6x+9y=159 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+9y+2y=159-38$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+2y=159-38$$

Tāpiri 9y ki te 2y.

$$11y=159-38$$

Tāpiri 159 ki te -38.

$$11y=121$$

Whakawehea ngā taha e rua ki te 11.

$$y=11$$

Whakaurua te 11 mō y ki 3x-y=19. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-11=19$$

Me tāpiri 11 ki ngā taha e rua o te whārite.

$$3x=30$$

Whakawehea ngā taha e rua ki te 3.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=11$$