Whakaoti i te {l}{3x+5y=-24}{-5x+2y=-22x}

$$\left. \begin{array} { l } { 3 x + 5 y = - 24 } \\ { - 5 x + 2 y = - 22 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{48}{79}\approx 0.607594937$$

$$y = -\frac{408}{79} = -5\frac{13}{79} \approx -5.164556962$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 22x ki ngā taha e rua.

$$-5x+2y+22x=0$$

Pahekotia te -5x me 22x, ka 17x.

$$17x+2y=0$$

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.

$$3x+5y=-24,17x+2y=0$$

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

$$3x+5y=-24$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y-24$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y-24\right)$$

Whakareatia \frac{1}{3} ki te -5y-24.

$$x=-\frac{5}{3}y-8$$

Whakakapia te -\frac{5y}{3}-8 mō te x ki tērā atu whārite, 17x+2y=0.

$$17\left(-\frac{5}{3}y-8\right)+2y=0$$

Whakareatia 17 ki te -\frac{5y}{3}-8.

$$-\frac{85}{3}y-136+2y=0$$

Tāpiri -\frac{85y}{3} ki te 2y.

$$-\frac{79}{3}y-136=0$$

Me tāpiri 136 ki ngā taha e rua o te whārite.

$$-\frac{79}{3}y=136$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{79}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{408}{79}$$

Whakaurua te -\frac{408}{79} mō y ki x=-\frac{5}{3}y-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\left(-\frac{408}{79}\right)-8$$

Whakareatia -\frac{5}{3} ki te -\frac{408}{79} 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.

$$x=\frac{680}{79}-8$$

Tāpiri -8 ki te \frac{680}{79}.

$$x=\frac{48}{79}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{48}{79},y=-\frac{408}{79}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 22x ki ngā taha e rua.

$$-5x+2y+22x=0$$

Pahekotia te -5x me 22x, ka 17x.

$$17x+2y=0$$

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

$$3x+5y=-24,17x+2y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\17&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\17&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\17&2\end{matrix}\right))\left(\begin{matrix}3&5\\17&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\17&2\end{matrix}\right))\left(\begin{matrix}-24\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\17&2\end{matrix}\right))\left(\begin{matrix}-24\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\17&2\end{matrix}\right))\left(\begin{matrix}-24\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-5\times 17}&-\frac{5}{3\times 2-5\times 17}\\-\frac{17}{3\times 2-5\times 17}&\frac{3}{3\times 2-5\times 17}\end{matrix}\right)\left(\begin{matrix}-24\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{79}&\frac{5}{79}\\\frac{17}{79}&-\frac{3}{79}\end{matrix}\right)\left(\begin{matrix}-24\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{79}\left(-24\right)\\\frac{17}{79}\left(-24\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{48}{79}\\-\frac{408}{79}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{48}{79},y=-\frac{408}{79}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 22x ki ngā taha e rua.

$$-5x+2y+22x=0$$

Pahekotia te -5x me 22x, ka 17x.

$$17x+2y=0$$

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.

$$3x+5y=-24,17x+2y=0$$

Kia ōrite ai a 3x me 17x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 17 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$17\times 3x+17\times 5y=17\left(-24\right),3\times 17x+3\times 2y=0$$

Whakarūnātia.

$$51x+85y=-408,51x+6y=0$$

Me tango 51x+6y=0 mai i 51x+85y=-408 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$51x-51x+85y-6y=-408$$

Tāpiri 51x ki te -51x. Ka whakakore atu ngā kupu 51x me -51x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$85y-6y=-408$$

Tāpiri 85y ki te -6y.

$$79y=-408$$

Whakawehea ngā taha e rua ki te 79.

$$y=-\frac{408}{79}$$

Whakaurua te -\frac{408}{79} mō y ki 17x+2y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$17x+2\left(-\frac{408}{79}\right)=0$$

Whakareatia 2 ki te -\frac{408}{79}.

$$17x-\frac{816}{79}=0$$

Me tāpiri \frac{816}{79} ki ngā taha e rua o te whārite.

$$17x=\frac{816}{79}$$

Whakawehea ngā taha e rua ki te 17.

$$x=\frac{48}{79}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{48}{79},y=-\frac{408}{79}$$