Whakaoti i te {l}{3(2x-5)+2y=-41}{x-3y/9-y=5}

$$\left. \begin{array} { l } { 3 ( 2 x - 5 ) + 2 y = - 41 } \\ { \frac { x - 3 y } { 9 } - y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-3$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2x-5.

$$6x-15+2y=-41$$

Me tāpiri te 15 ki ngā taha e rua.

$$6x+2y=-41+15$$

Tāpirihia te -41 ki te 15, ka -26.

$$6x+2y=-26$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 9.

$$x-3y-9y=45$$

Pahekotia te -3y me -9y, ka -12y.

$$x-12y=45$$

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.

$$6x+2y=-26,x-12y=45$$

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

$$6x+2y=-26$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$6x=-2y-26$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{1}{6}\left(-2y-26\right)$$

Whakareatia \frac{1}{6} ki te -2y-26.

$$x=-\frac{1}{3}y-\frac{13}{3}$$

Whakakapia te \frac{-y-13}{3} mō te x ki tērā atu whārite, x-12y=45.

$$-\frac{1}{3}y-\frac{13}{3}-12y=45$$

Tāpiri -\frac{y}{3} ki te -12y.

$$-\frac{37}{3}y-\frac{13}{3}=45$$

Me tāpiri \frac{13}{3} ki ngā taha e rua o te whārite.

$$-\frac{37}{3}y=\frac{148}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{37}{3}, 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{1}{3}y-\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\left(-4\right)-\frac{13}{3}$$

Whakareatia -\frac{1}{3} ki te -4.

$$x=\frac{4-13}{3}$$

Tāpiri -\frac{13}{3} ki te \frac{4}{3} 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.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2x-5.

$$6x-15+2y=-41$$

Me tāpiri te 15 ki ngā taha e rua.

$$6x+2y=-41+15$$

Tāpirihia te -41 ki te 15, ka -26.

$$6x+2y=-26$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 9.

$$x-3y-9y=45$$

Pahekotia te -3y me -9y, ka -12y.

$$x-12y=45$$

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

$$6x+2y=-26,x-12y=45$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}6&2\\1&-12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-26\\45\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}6&2\\1&-12\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&2\\1&-12\end{matrix}\right))\left(\begin{matrix}6&2\\1&-12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\1&-12\end{matrix}\right))\left(\begin{matrix}-26\\45\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\1&-12\end{matrix}\right))\left(\begin{matrix}-26\\45\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&2\\1&-12\end{matrix}\right))\left(\begin{matrix}-26\\45\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{12}{6\left(-12\right)-2}&-\frac{2}{6\left(-12\right)-2}\\-\frac{1}{6\left(-12\right)-2}&\frac{6}{6\left(-12\right)-2}\end{matrix}\right)\left(\begin{matrix}-26\\45\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{37}&\frac{1}{37}\\\frac{1}{74}&-\frac{3}{37}\end{matrix}\right)\left(\begin{matrix}-26\\45\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{37}\left(-26\right)+\frac{1}{37}\times 45\\\frac{1}{74}\left(-26\right)-\frac{3}{37}\times 45\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-3,y=-4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2x-5.

$$6x-15+2y=-41$$

Me tāpiri te 15 ki ngā taha e rua.

$$6x+2y=-41+15$$

Tāpirihia te -41 ki te 15, ka -26.

$$6x+2y=-26$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 9.

$$x-3y-9y=45$$

Pahekotia te -3y me -9y, ka -12y.

$$x-12y=45$$

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.

$$6x+2y=-26,x-12y=45$$

Kia ōrite ai a 6x 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 6.

$$6x+2y=-26,6x+6\left(-12\right)y=6\times 45$$

Whakarūnātia.

$$6x+2y=-26,6x-72y=270$$

Me tango 6x-72y=270 mai i 6x+2y=-26 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+2y+72y=-26-270$$

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.

$$2y+72y=-26-270$$

Tāpiri 2y ki te 72y.

$$74y=-26-270$$

Tāpiri -26 ki te -270.

$$74y=-296$$

Whakawehea ngā taha e rua ki te 74.

$$y=-4$$

Whakaurua te -4 mō y ki x-12y=45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-12\left(-4\right)=45$$

Whakareatia -12 ki te -4.

$$x+48=45$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$x=-3$$

Kua oti te pūnaha te whakatau.

$$x=-3,y=-4$$