Whakaoti i te {l}{30x+15y=675}{42x+20y=940}

$$\left\{ \begin{array} { l } { 30 x + 15 y = 675 } \\ { 42 x + 20 y = 940 } \end{array} \right.$$

Whakaoti mō x, y

$$x=20$$

$$y=5$$

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.

$$30x+15y=675,42x+20y=940$$

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

$$30x+15y=675$$

Me tango 15y mai i ngā taha e rua o te whārite.

$$30x=-15y+675$$

Whakawehea ngā taha e rua ki te 30.

$$x=\frac{1}{30}\left(-15y+675\right)$$

Whakareatia \frac{1}{30} ki te -15y+675.

$$x=-\frac{1}{2}y+\frac{45}{2}$$

Whakakapia te \frac{-y+45}{2} mō te x ki tērā atu whārite, 42x+20y=940.

$$42\left(-\frac{1}{2}y+\frac{45}{2}\right)+20y=940$$

Whakareatia 42 ki te \frac{-y+45}{2}.

$$-21y+945+20y=940$$

Tāpiri -21y ki te 20y.

$$-y+945=940$$

Me tango 945 mai i ngā taha e rua o te whārite.

$$-y=-5$$

Whakawehea ngā taha e rua ki te -1.

$$y=5$$

Whakaurua te 5 mō y ki x=-\frac{1}{2}y+\frac{45}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times 5+\frac{45}{2}$$

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

$$x=\frac{-5+45}{2}$$

Tāpiri \frac{45}{2} ki te -\frac{5}{2} 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.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=5$$

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.

$$30x+15y=675,42x+20y=940$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}30&15\\42&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}675\\940\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}30&15\\42&20\end{matrix}\right).

$$inverse(\left(\begin{matrix}30&15\\42&20\end{matrix}\right))\left(\begin{matrix}30&15\\42&20\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&15\\42&20\end{matrix}\right))\left(\begin{matrix}675\\940\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&15\\42&20\end{matrix}\right))\left(\begin{matrix}675\\940\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&15\\42&20\end{matrix}\right))\left(\begin{matrix}675\\940\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{20}{30\times 20-15\times 42}&-\frac{15}{30\times 20-15\times 42}\\-\frac{42}{30\times 20-15\times 42}&\frac{30}{30\times 20-15\times 42}\end{matrix}\right)\left(\begin{matrix}675\\940\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}&\frac{1}{2}\\\frac{7}{5}&-1\end{matrix}\right)\left(\begin{matrix}675\\940\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{3}\times 675+\frac{1}{2}\times 940\\\frac{7}{5}\times 675-940\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=20,y=5$$

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.

$$30x+15y=675,42x+20y=940$$

Kia ōrite ai a 30x me 42x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 42 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 30.

$$42\times 30x+42\times 15y=42\times 675,30\times 42x+30\times 20y=30\times 940$$

Whakarūnātia.

$$1260x+630y=28350,1260x+600y=28200$$

Me tango 1260x+600y=28200 mai i 1260x+630y=28350 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$1260x-1260x+630y-600y=28350-28200$$

Tāpiri 1260x ki te -1260x. Ka whakakore atu ngā kupu 1260x me -1260x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$630y-600y=28350-28200$$

Tāpiri 630y ki te -600y.

$$30y=28350-28200$$

Tāpiri 28350 ki te -28200.

$$30y=150$$

Whakawehea ngā taha e rua ki te 30.

$$y=5$$

Whakaurua te 5 mō y ki 42x+20y=940. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$42x+20\times 5=940$$

Whakareatia 20 ki te 5.

$$42x+100=940$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$42x=840$$

Whakawehea ngā taha e rua ki te 42.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=5$$