Whakaoti i te {l}{2x+y-7=0}{17x-11y-8=0}

$$\left. \begin{array} { l } { 2 x + y - 7 = 0 } \\ { 17 x - 11 y - 8 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{85}{39} = 2\frac{7}{39} \approx 2.179487179$$

$$y = \frac{103}{39} = 2\frac{25}{39} \approx 2.641025641$$

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.

$$2x+y-7=0,17x-11y-8=0$$

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

$$2x+y-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2x+y=7$$

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

$$2x=-y+7$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+7\right)$$

Whakareatia \frac{1}{2} ki te -y+7.

$$x=-\frac{1}{2}y+\frac{7}{2}$$

Whakakapia te \frac{-y+7}{2} mō te x ki tērā atu whārite, 17x-11y-8=0.

$$17\left(-\frac{1}{2}y+\frac{7}{2}\right)-11y-8=0$$

Whakareatia 17 ki te \frac{-y+7}{2}.

$$-\frac{17}{2}y+\frac{119}{2}-11y-8=0$$

Tāpiri -\frac{17y}{2} ki te -11y.

$$-\frac{39}{2}y+\frac{119}{2}-8=0$$

Tāpiri \frac{119}{2} ki te -8.

$$-\frac{39}{2}y+\frac{103}{2}=0$$

Me tango \frac{103}{2} mai i ngā taha e rua o te whārite.

$$-\frac{39}{2}y=-\frac{103}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{39}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{103}{39}$$

Whakaurua te \frac{103}{39} mō y ki x=-\frac{1}{2}y+\frac{7}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{103}{39}+\frac{7}{2}$$

Whakareatia -\frac{1}{2} ki te \frac{103}{39} 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.

$$x=-\frac{103}{78}+\frac{7}{2}$$

Tāpiri \frac{7}{2} ki te -\frac{103}{78} 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.

$$x=\frac{85}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{85}{39},y=\frac{103}{39}$$

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.

$$2x+y-7=0,17x-11y-8=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\17&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&1\\17&-11\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&1\\17&-11\end{matrix}\right))\left(\begin{matrix}2&1\\17&-11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\17&-11\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\17&-11\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\17&-11\end{matrix}\right))\left(\begin{matrix}7\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{2\left(-11\right)-17}&-\frac{1}{2\left(-11\right)-17}\\-\frac{17}{2\left(-11\right)-17}&\frac{2}{2\left(-11\right)-17}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{39}&\frac{1}{39}\\\frac{17}{39}&-\frac{2}{39}\end{matrix}\right)\left(\begin{matrix}7\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{39}\times 7+\frac{1}{39}\times 8\\\frac{17}{39}\times 7-\frac{2}{39}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{85}{39}\\\frac{103}{39}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{85}{39},y=\frac{103}{39}$$

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.

$$2x+y-7=0,17x-11y-8=0$$

Kia ōrite ai a 2x 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 2.

$$17\times 2x+17y+17\left(-7\right)=0,2\times 17x+2\left(-11\right)y+2\left(-8\right)=0$$

Whakarūnātia.

$$34x+17y-119=0,34x-22y-16=0$$

Me tango 34x-22y-16=0 mai i 34x+17y-119=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$34x-34x+17y+22y-119+16=0$$

Tāpiri 34x ki te -34x. Ka whakakore atu ngā kupu 34x me -34x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$17y+22y-119+16=0$$

Tāpiri 17y ki te 22y.

$$39y-119+16=0$$

Tāpiri -119 ki te 16.

$$39y-103=0$$

Me tāpiri 103 ki ngā taha e rua o te whārite.

$$39y=103$$

Whakawehea ngā taha e rua ki te 39.

$$y=\frac{103}{39}$$

Whakaurua te \frac{103}{39} mō y ki 17x-11y-8=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$17x-11\times \frac{103}{39}-8=0$$

Whakareatia -11 ki te \frac{103}{39}.

$$17x-\frac{1133}{39}-8=0$$

Tāpiri -\frac{1133}{39} ki te -8.

$$17x-\frac{1445}{39}=0$$

Me tāpiri \frac{1445}{39} ki ngā taha e rua o te whārite.

$$17x=\frac{1445}{39}$$

Whakawehea ngā taha e rua ki te 17.

$$x=\frac{85}{39}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{85}{39},y=\frac{103}{39}$$