Whakaoti i te {l}{y=x+300}{0.09x+0.07y=365}

$$\left. \begin{array} { l } { y = x + 300 } \\ { 0.09 x + 0.07 y = 365 } \end{array} \right.$$

Whakaoti mō y, x

$$x=2150$$

$$y=2450$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te x mai i ngā taha e rua.

$$y-x=300$$

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.

$$y-x=300,0.07y+0.09x=365$$

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

$$y-x=300$$

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

$$y=x+300$$

Whakakapia te x+300 mō te y ki tērā atu whārite, 0.07y+0.09x=365.

$$0.07\left(x+300\right)+0.09x=365$$

Whakareatia 0.07 ki te x+300.

$$0.07x+21+0.09x=365$$

Tāpiri \frac{7x}{100} ki te \frac{9x}{100}.

$$0.16x+21=365$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$0.16x=344$$

Whakawehea ngā taha e rua o te whārite ki te 0.16, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=2150$$

Whakaurua te 2150 mō x ki y=x+300. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=2150+300$$

Tāpiri 300 ki te 2150.

$$y=2450$$

Kua oti te pūnaha te whakatau.

$$y=2450,x=2150$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te x mai i ngā taha e rua.

$$y-x=300$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$y-x=300,0.07y+0.09x=365$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}300\\365\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right))\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right))\left(\begin{matrix}300\\365\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right))\left(\begin{matrix}300\\365\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\0.07&0.09\end{matrix}\right))\left(\begin{matrix}300\\365\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{0.09}{0.09-\left(-0.07\right)}&-\frac{-1}{0.09-\left(-0.07\right)}\\-\frac{0.07}{0.09-\left(-0.07\right)}&\frac{1}{0.09-\left(-0.07\right)}\end{matrix}\right)\left(\begin{matrix}300\\365\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0.5625&6.25\\-0.4375&6.25\end{matrix}\right)\left(\begin{matrix}300\\365\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0.5625\times 300+6.25\times 365\\-0.4375\times 300+6.25\times 365\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}2450\\2150\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=2450,x=2150$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te x mai i ngā taha e rua.

$$y-x=300$$

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.

$$y-x=300,0.07y+0.09x=365$$

Kia ōrite ai a y me \frac{7y}{100}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.07 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$0.07y+0.07\left(-1\right)x=0.07\times 300,0.07y+0.09x=365$$

Whakarūnātia.

$$0.07y-0.07x=21,0.07y+0.09x=365$$

Me tango 0.07y+0.09x=365 mai i 0.07y-0.07x=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.07y-0.07y-0.07x-0.09x=21-365$$

Tāpiri \frac{7y}{100} ki te -\frac{7y}{100}. Ka whakakore atu ngā kupu \frac{7y}{100} me -\frac{7y}{100}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-0.07x-0.09x=21-365$$

Tāpiri -\frac{7x}{100} ki te -\frac{9x}{100}.

$$-0.16x=21-365$$

Tāpiri 21 ki te -365.

$$-0.16x=-344$$

Whakawehea ngā taha e rua o te whārite ki te -0.16, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=2150$$

Whakaurua te 2150 mō x ki 0.07y+0.09x=365. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$0.07y+0.09\times 2150=365$$

Whakareatia 0.09 ki te 2150.

$$0.07y+193.5=365$$

Me tango 193.5 mai i ngā taha e rua o te whārite.

$$0.07y=171.5$$

Whakawehea ngā taha e rua o te whārite ki te 0.07, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2450$$

Kua oti te pūnaha te whakatau.

$$y=2450,x=2150$$