Whakaoti i te {l}{2x+3y=10}{4x+5y=42}

$$\left. \begin{array} { l } { 2 x + 3 y = 10 } \\ { 4 x + 5 y = 42 } \end{array} \right.$$

Whakaoti mō x, y

$$x=38$$

$$y=-22$$

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.

$$2x+3y=10,4x+5y=42$$

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

$$2x+3y=10$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+10\right)$$

Whakareatia \frac{1}{2} ki te -3y+10.

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

Whakakapia te -\frac{3y}{2}+5 mō te x ki tērā atu whārite, 4x+5y=42.

$$4\left(-\frac{3}{2}y+5\right)+5y=42$$

Whakareatia 4 ki te -\frac{3y}{2}+5.

$$-6y+20+5y=42$$

Tāpiri -6y ki te 5y.

$$-y+20=42$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-y=22$$

Whakawehea ngā taha e rua ki te -1.

$$y=-22$$

Whakaurua te -22 mō y ki x=-\frac{3}{2}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\left(-22\right)+5$$

Whakareatia -\frac{3}{2} ki te -22.

$$x=33+5$$

Tāpiri 5 ki te 33.

$$x=38$$

Kua oti te pūnaha te whakatau.

$$x=38,y=-22$$

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.

$$2x+3y=10,4x+5y=42$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\42\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\4&5\end{matrix}\right))\left(\begin{matrix}2&3\\4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&5\end{matrix}\right))\left(\begin{matrix}10\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&5\end{matrix}\right))\left(\begin{matrix}10\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&5\end{matrix}\right))\left(\begin{matrix}10\\42\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-3\times 4}&-\frac{3}{2\times 5-3\times 4}\\-\frac{4}{2\times 5-3\times 4}&\frac{2}{2\times 5-3\times 4}\end{matrix}\right)\left(\begin{matrix}10\\42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}&\frac{3}{2}\\2&-1\end{matrix}\right)\left(\begin{matrix}10\\42\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\times 10+\frac{3}{2}\times 42\\2\times 10-42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}38\\-22\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=38,y=-22$$

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.

$$2x+3y=10,4x+5y=42$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\times 3y=4\times 10,2\times 4x+2\times 5y=2\times 42$$

Whakarūnātia.

$$8x+12y=40,8x+10y=84$$

Me tango 8x+10y=84 mai i 8x+12y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+12y-10y=40-84$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-10y=40-84$$

Tāpiri 12y ki te -10y.

$$2y=40-84$$

Tāpiri 40 ki te -84.

$$2y=-44$$

Whakawehea ngā taha e rua ki te 2.

$$y=-22$$

Whakaurua te -22 mō y ki 4x+5y=42. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+5\left(-22\right)=42$$

Whakareatia 5 ki te -22.

$$4x-110=42$$

Me tāpiri 110 ki ngā taha e rua o te whārite.

$$4x=152$$

Whakawehea ngā taha e rua ki te 4.

$$x=38$$

Kua oti te pūnaha te whakatau.

$$x=38,y=-22$$