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.

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