Whakaoti i te {l}{text{(1)}-5x+2y=15}{text{(I)}x+3y=32}

$$\left. \begin{array} { l } { \text { (1) } - 5 x + 2 y = 15 } \\ { \text { (I) } x + 3 y = 32 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{19}{17} = 1\frac{2}{17} \approx 1.117647059$$

$$y = \frac{175}{17} = 10\frac{5}{17} \approx 10.294117647$$

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.

$$-5x+2y=15,x+3y=32$$

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

$$-5x+2y=15$$

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

$$-5x=-2y+15$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-2y+15\right)$$

Whakareatia -\frac{1}{5} ki te -2y+15.

$$x=\frac{2}{5}y-3$$

Whakakapia te \frac{2y}{5}-3 mō te x ki tērā atu whārite, x+3y=32.

$$\frac{2}{5}y-3+3y=32$$

Tāpiri \frac{2y}{5} ki te 3y.

$$\frac{17}{5}y-3=32$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$\frac{17}{5}y=35$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{175}{17}$$

Whakaurua te \frac{175}{17} mō y ki x=\frac{2}{5}y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times \frac{175}{17}-3$$

Whakareatia \frac{2}{5} ki te \frac{175}{17} 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.

$$x=\frac{70}{17}-3$$

Tāpiri -3 ki te \frac{70}{17}.

$$x=\frac{19}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{17},y=\frac{175}{17}$$

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.

$$-5x+2y=15,x+3y=32$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&2\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\32\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-5&2\\1&3\end{matrix}\right))\left(\begin{matrix}-5&2\\1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&2\\1&3\end{matrix}\right))\left(\begin{matrix}15\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&2\\1&3\end{matrix}\right))\left(\begin{matrix}15\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&2\\1&3\end{matrix}\right))\left(\begin{matrix}15\\32\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-5\times 3-2}&-\frac{2}{-5\times 3-2}\\-\frac{1}{-5\times 3-2}&-\frac{5}{-5\times 3-2}\end{matrix}\right)\left(\begin{matrix}15\\32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{17}&\frac{2}{17}\\\frac{1}{17}&\frac{5}{17}\end{matrix}\right)\left(\begin{matrix}15\\32\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{17}\times 15+\frac{2}{17}\times 32\\\frac{1}{17}\times 15+\frac{5}{17}\times 32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{17}\\\frac{175}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19}{17},y=\frac{175}{17}$$

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.

$$-5x+2y=15,x+3y=32$$

Kia ōrite ai a -5x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$-5x+2y=15,-5x-5\times 3y=-5\times 32$$

Whakarūnātia.

$$-5x+2y=15,-5x-15y=-160$$

Me tango -5x-15y=-160 mai i -5x+2y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x+2y+15y=15+160$$

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

$$2y+15y=15+160$$

Tāpiri 2y ki te 15y.

$$17y=15+160$$

Tāpiri 15 ki te 160.

$$17y=175$$

Whakawehea ngā taha e rua ki te 17.

$$y=\frac{175}{17}$$

Whakaurua te \frac{175}{17} mō y ki x+3y=32. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3\times \frac{175}{17}=32$$

Whakareatia 3 ki te \frac{175}{17}.

$$x+\frac{525}{17}=32$$

Me tango \frac{525}{17} mai i ngā taha e rua o te whārite.

$$x=\frac{19}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{17},y=\frac{175}{17}$$