Whakaoti i te {l}{3x+2y=17}{5x-y=2}

$$\left\{ \begin{array} { l } { 3 x + 2 y = 17 } \\ { 5 x - y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{21}{13} = 1\frac{8}{13} \approx 1.615384615$$

$$y = \frac{79}{13} = 6\frac{1}{13} \approx 6.076923077$$

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.

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

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

$$3x+2y=17$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$3x=-2y+17$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-2y+17\right)$$

Whakareatia \frac{1}{3} ki te -2y+17.

$$x=-\frac{2}{3}y+\frac{17}{3}$$

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

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

Whakareatia 5 ki te \frac{-2y+17}{3}.

$$-\frac{10}{3}y+\frac{85}{3}-y=2$$

Tāpiri -\frac{10y}{3} ki te -y.

$$-\frac{13}{3}y+\frac{85}{3}=2$$

Me tango \frac{85}{3} mai i ngā taha e rua o te whārite.

$$-\frac{13}{3}y=-\frac{79}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{13}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{79}{13}$$

Whakaurua te \frac{79}{13} mō y ki x=-\frac{2}{3}y+\frac{17}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{3}\times \frac{79}{13}+\frac{17}{3}$$

Whakareatia -\frac{2}{3} ki te \frac{79}{13} 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.

$$x=-\frac{158}{39}+\frac{17}{3}$$

Tāpiri \frac{17}{3} ki te -\frac{158}{39} 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.

$$x=\frac{21}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{21}{13},y=\frac{79}{13}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&2\\5&-1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}&\frac{2}{13}\\\frac{5}{13}&-\frac{3}{13}\end{matrix}\right)\left(\begin{matrix}17\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\times 17+\frac{2}{13}\times 2\\\frac{5}{13}\times 17-\frac{3}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{21}{13}\\\frac{79}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{21}{13},y=\frac{79}{13}$$

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.

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

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5\times 2y=5\times 17,3\times 5x+3\left(-1\right)y=3\times 2$$

Whakarūnātia.

$$15x+10y=85,15x-3y=6$$

Me tango 15x-3y=6 mai i 15x+10y=85 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+10y+3y=85-6$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y+3y=85-6$$

Tāpiri 10y ki te 3y.

$$13y=85-6$$

Tāpiri 85 ki te -6.

$$13y=79$$

Whakawehea ngā taha e rua ki te 13.

$$y=\frac{79}{13}$$

Whakaurua te \frac{79}{13} mō y ki 5x-y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-\frac{79}{13}=2$$

Me tāpiri \frac{79}{13} ki ngā taha e rua o te whārite.

$$5x=\frac{105}{13}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{21}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{21}{13},y=\frac{79}{13}$$