Whakaoti i te {l}{Cx+y=69}{2x+y=87}

$$\left. \begin{array} { l } { C x + y = 69 } \\ { 2 x + y = 87 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{18}{C-2}$$

$$y=-\frac{3\left(46-29C\right)}{C-2}$$

$$C\neq 2$$

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.

$$Cx+y=69,2x+y=87$$

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

$$Cx+y=69$$

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

$$Cx=-y+69$$

Whakawehea ngā taha e rua ki te C.

$$x=\frac{1}{C}\left(-y+69\right)$$

Whakareatia \frac{1}{C} ki te -y+69.

$$x=\left(-\frac{1}{C}\right)y+\frac{69}{C}$$

Whakakapia te \frac{69-y}{C} mō te x ki tērā atu whārite, 2x+y=87.

$$2\left(\left(-\frac{1}{C}\right)y+\frac{69}{C}\right)+y=87$$

Whakareatia 2 ki te \frac{69-y}{C}.

$$\left(-\frac{2}{C}\right)y+\frac{138}{C}+y=87$$

Tāpiri -\frac{2y}{C} ki te y.

$$\frac{C-2}{C}y+\frac{138}{C}=87$$

Me tango \frac{138}{C} mai i ngā taha e rua o te whārite.

$$\frac{C-2}{C}y=87-\frac{138}{C}$$

Whakawehea ngā taha e rua ki te \frac{-2+C}{C}.

$$y=\frac{3\left(29C-46\right)}{C-2}$$

Whakaurua te \frac{3\left(-46+29C\right)}{-2+C} mō y ki x=\left(-\frac{1}{C}\right)y+\frac{69}{C}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\left(-\frac{1}{C}\right)\times \frac{3\left(29C-46\right)}{C-2}+\frac{69}{C}$$

Whakareatia -\frac{1}{C} ki te \frac{3\left(-46+29C\right)}{-2+C}.

$$x=-\frac{3\left(29C-46\right)}{C\left(C-2\right)}+\frac{69}{C}$$

Tāpiri \frac{69}{C} ki te -\frac{3\left(-46+29C\right)}{C\left(-2+C\right)}.

$$x=-\frac{18}{C-2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{18}{C-2},y=\frac{3\left(29C-46\right)}{C-2}$$

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.

$$Cx+y=69,2x+y=87$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}C&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}C&1\\2&1\end{matrix}\right))\left(\begin{matrix}C&1\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}C&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}C&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}C&1\\2&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{C-2}&-\frac{1}{C-2}\\-\frac{2}{C-2}&\frac{C}{C-2}\end{matrix}\right)\left(\begin{matrix}69\\87\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{C-2}\times 69+\left(-\frac{1}{C-2}\right)\times 87\\\left(-\frac{2}{C-2}\right)\times 69+\frac{C}{C-2}\times 87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{18}{C-2}\\\frac{3\left(29C-46\right)}{C-2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{18}{C-2},y=\frac{3\left(29C-46\right)}{C-2}$$

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.

$$Cx+y=69,2x+y=87$$

Me tango 2x+y=87 mai i Cx+y=69 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$Cx-2x+y-y=69-87$$

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

$$Cx-2x=69-87$$

Tāpiri Cx ki te -2x.

$$\left(C-2\right)x=69-87$$

Tāpiri 69 ki te -87.

$$\left(C-2\right)x=-18$$

Whakawehea ngā taha e rua ki te C-2.

$$x=-\frac{18}{C-2}$$

Whakaurua te -\frac{18}{C-2} mō x ki 2x+y=87. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\left(-\frac{18}{C-2}\right)+y=87$$

Whakareatia 2 ki te -\frac{18}{C-2}.

$$-\frac{36}{C-2}+y=87$$

Me tāpiri \frac{36}{C-2} ki ngā taha e rua o te whārite.

$$y=\frac{3\left(29C-46\right)}{C-2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{18}{C-2},y=\frac{3\left(29C-46\right)}{C-2}$$