Whakaoti i te {l}{2x-3y-10=0}{17y=-11-3x}

$$\left. \begin{array} { l } { 2 x - 3 y - 10 = 0 } \\ { 17 y = - 11 - 3 x } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{137}{43} = 3\frac{8}{43} \approx 3.186046512$$

$$y = -\frac{52}{43} = -1\frac{9}{43} \approx -1.209302326$$

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 3x ki ngā taha e rua.

$$17y+3x=-11$$

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.

$$2x-3y=10,3x+17y=-11$$

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, 3x+17y=-11.

$$3\left(\frac{3}{2}y+5\right)+17y=-11$$

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

$$\frac{9}{2}y+15+17y=-11$$

Tāpiri \frac{9y}{2} ki te 17y.

$$\frac{43}{2}y+15=-11$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$\frac{43}{2}y=-26$$

Whakawehea ngā taha e rua o te whārite ki te \frac{43}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{52}{43}$$

Whakaurua te -\frac{52}{43} 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.

$$x=\frac{3}{2}\left(-\frac{52}{43}\right)+5$$

Whakareatia \frac{3}{2} ki te -\frac{52}{43} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{78}{43}+5$$

Tāpiri 5 ki te -\frac{78}{43}.

$$x=\frac{137}{43}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{137}{43},y=-\frac{52}{43}$$

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 3x ki ngā taha e rua.

$$17y+3x=-11$$

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

$$2x-3y=10,3x+17y=-11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\3&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\3&17\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\3&17\end{matrix}\right))\left(\begin{matrix}2&-3\\3&17\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&17\end{matrix}\right))\left(\begin{matrix}10\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&17\end{matrix}\right))\left(\begin{matrix}10\\-11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&17\end{matrix}\right))\left(\begin{matrix}10\\-11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{2\times 17-\left(-3\times 3\right)}&-\frac{-3}{2\times 17-\left(-3\times 3\right)}\\-\frac{3}{2\times 17-\left(-3\times 3\right)}&\frac{2}{2\times 17-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}10\\-11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{43}&\frac{3}{43}\\-\frac{3}{43}&\frac{2}{43}\end{matrix}\right)\left(\begin{matrix}10\\-11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{17}{43}\times 10+\frac{3}{43}\left(-11\right)\\-\frac{3}{43}\times 10+\frac{2}{43}\left(-11\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{137}{43}\\-\frac{52}{43}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{137}{43},y=-\frac{52}{43}$$

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 3x ki ngā taha e rua.

$$17y+3x=-11$$

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.

$$2x-3y=10,3x+17y=-11$$

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.

$$3\times 2x+3\left(-3\right)y=3\times 10,2\times 3x+2\times 17y=2\left(-11\right)$$

Whakarūnātia.

$$6x-9y=30,6x+34y=-22$$

Me tango 6x+34y=-22 mai i 6x-9y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-9y-34y=30+22$$

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.

$$-9y-34y=30+22$$

Tāpiri -9y ki te -34y.

$$-43y=30+22$$

Tāpiri 30 ki te 22.

$$-43y=52$$

Whakawehea ngā taha e rua ki te -43.

$$y=-\frac{52}{43}$$

Whakaurua te -\frac{52}{43} mō y ki 3x+17y=-11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+17\left(-\frac{52}{43}\right)=-11$$

Whakareatia 17 ki te -\frac{52}{43}.

$$3x-\frac{884}{43}=-11$$

Me tāpiri \frac{884}{43} ki ngā taha e rua o te whārite.

$$3x=\frac{411}{43}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{137}{43}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{137}{43},y=-\frac{52}{43}$$