Whakaoti i te {l}{x+y=17}{2.6x+3.5y=55}

$$\left. \begin{array} { l } { x + y = 17 } \\ { 2.6 x + 3.5 y = 55 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=12$$

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.

$$x+y=17,2.6x+3.5y=55$$

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

$$x+y=17$$

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

$$x=-y+17$$

Whakakapia te -y+17 mō te x ki tērā atu whārite, 2.6x+3.5y=55.

$$2.6\left(-y+17\right)+3.5y=55$$

Whakareatia 2.6 ki te -y+17.

$$-2.6y+44.2+3.5y=55$$

Tāpiri -\frac{13y}{5} ki te \frac{7y}{2}.

$$0.9y+44.2=55$$

Me tango 44.2 mai i ngā taha e rua o te whārite.

$$0.9y=10.8$$

Whakawehea ngā taha e rua o te whārite ki te 0.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=12$$

Whakaurua te 12 mō y ki x=-y+17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-12+17$$

Tāpiri 17 ki te -12.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=12$$

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.

$$x+y=17,2.6x+3.5y=55$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\55\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right))\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right))\left(\begin{matrix}17\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right))\left(\begin{matrix}17\\55\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\2.6&3.5\end{matrix}\right))\left(\begin{matrix}17\\55\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3.5}{3.5-2.6}&-\frac{1}{3.5-2.6}\\-\frac{2.6}{3.5-2.6}&\frac{1}{3.5-2.6}\end{matrix}\right)\left(\begin{matrix}17\\55\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{9}&-\frac{10}{9}\\-\frac{26}{9}&\frac{10}{9}\end{matrix}\right)\left(\begin{matrix}17\\55\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{35}{9}\times 17-\frac{10}{9}\times 55\\-\frac{26}{9}\times 17+\frac{10}{9}\times 55\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=12$$

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.

$$x+y=17,2.6x+3.5y=55$$

Kia ōrite ai a x me \frac{13x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2.6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2.6x+2.6y=2.6\times 17,2.6x+3.5y=55$$

Whakarūnātia.

$$2.6x+2.6y=44.2,2.6x+3.5y=55$$

Me tango 2.6x+3.5y=55 mai i 2.6x+2.6y=44.2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2.6x-2.6x+2.6y-3.5y=44.2-55$$

Tāpiri \frac{13x}{5} ki te -\frac{13x}{5}. Ka whakakore atu ngā kupu \frac{13x}{5} me -\frac{13x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2.6y-3.5y=44.2-55$$

Tāpiri \frac{13y}{5} ki te -\frac{7y}{2}.

$$-0.9y=44.2-55$$

Tāpiri 44.2 ki te -55.

$$-0.9y=-10.8$$

Whakawehea ngā taha e rua o te whārite ki te -0.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=12$$

Whakaurua te 12 mō y ki 2.6x+3.5y=55. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2.6x+3.5\times 12=55$$

Whakareatia 3.5 ki te 12.

$$2.6x+42=55$$

Me tango 42 mai i ngā taha e rua o te whārite.

$$2.6x=13$$

Whakawehea ngā taha e rua o te whārite ki te 2.6, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=12$$