Whakaoti i te {l}{2x-3y-10=0}{7y=-17-8x}

$$\left\{ \begin{array} { l } { 2 x - 3 y - 10 = 0 } \\ { 7 y = - 17 - 8 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-3y=10$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$7y+8x=-17$$

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.

$$2x-3y=10,8x+7y=-17$$

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

$$2x-3y=10$$

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

$$2x=3y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+10\right)$$

Whakareatia \frac{1}{2} ki te 3y+10.

$$x=\frac{3}{2}y+5$$

Whakakapia te \frac{3y}{2}+5 mō te x ki tērā atu whārite, 8x+7y=-17.

$$8\left(\frac{3}{2}y+5\right)+7y=-17$$

Whakareatia 8 ki te \frac{3y}{2}+5.

$$12y+40+7y=-17$$

Tāpiri 12y ki te 7y.

$$19y+40=-17$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$19y=-57$$

Whakawehea ngā taha e rua ki te 19.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{3}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-3\right)+5$$

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

$$x=-\frac{9}{2}+5$$

Tāpiri 5 ki te -\frac{9}{2}.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-3y=10$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$7y+8x=-17$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$2x-3y=10,8x+7y=-17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\8&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\8&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\8&7\end{matrix}\right))\left(\begin{matrix}2&-3\\8&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\8&7\end{matrix}\right))\left(\begin{matrix}10\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\8&7\end{matrix}\right))\left(\begin{matrix}10\\-17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\8&7\end{matrix}\right))\left(\begin{matrix}10\\-17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-\left(-3\times 8\right)}&-\frac{-3}{2\times 7-\left(-3\times 8\right)}\\-\frac{8}{2\times 7-\left(-3\times 8\right)}&\frac{2}{2\times 7-\left(-3\times 8\right)}\end{matrix}\right)\left(\begin{matrix}10\\-17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{38}&\frac{3}{38}\\-\frac{4}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}10\\-17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{38}\times 10+\frac{3}{38}\left(-17\right)\\-\frac{4}{19}\times 10+\frac{1}{19}\left(-17\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 10 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-3y=10$$

Whakaarohia te whārite tuarua. Me tāpiri te 8x ki ngā taha e rua.

$$7y+8x=-17$$

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.

$$2x-3y=10,8x+7y=-17$$

Kia ōrite ai a 2x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$8\times 2x+8\left(-3\right)y=8\times 10,2\times 8x+2\times 7y=2\left(-17\right)$$

Whakarūnātia.

$$16x-24y=80,16x+14y=-34$$

Me tango 16x+14y=-34 mai i 16x-24y=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$16x-16x-24y-14y=80+34$$

Tāpiri 16x ki te -16x. Ka whakakore atu ngā kupu 16x me -16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-24y-14y=80+34$$

Tāpiri -24y ki te -14y.

$$-38y=80+34$$

Tāpiri 80 ki te 34.

$$-38y=114$$

Whakawehea ngā taha e rua ki te -38.

$$y=-3$$

Whakaurua te -3 mō y ki 8x+7y=-17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x+7\left(-3\right)=-17$$

Whakareatia 7 ki te -3.

$$8x-21=-17$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$8x=4$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=-3$$