Whakaoti i te {l}{5x+3y=450.000}{3x+4y=913.000}

$$\left. \begin{array} { l } { 5 x + 3 y = 450.000 } \\ { 3 x + 4 y = 913.000 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{939}{11} = -85\frac{4}{11} \approx -85.363636364$$

$$y = \frac{3215}{11} = 292\frac{3}{11} \approx 292.272727273$$

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.

$$5x+3y=450,3x+4y=913$$

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

$$5x+3y=450$$

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

$$5x=-3y+450$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+450\right)$$

Whakareatia \frac{1}{5} ki te -3y+450.

$$x=-\frac{3}{5}y+90$$

Whakakapia te -\frac{3y}{5}+90 mō te x ki tērā atu whārite, 3x+4y=913.

$$3\left(-\frac{3}{5}y+90\right)+4y=913$$

Whakareatia 3 ki te -\frac{3y}{5}+90.

$$-\frac{9}{5}y+270+4y=913$$

Tāpiri -\frac{9y}{5} ki te 4y.

$$\frac{11}{5}y+270=913$$

Me tango 270 mai i ngā taha e rua o te whārite.

$$\frac{11}{5}y=643$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3215}{11}$$

Whakaurua te \frac{3215}{11} mō y ki x=-\frac{3}{5}y+90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\times \frac{3215}{11}+90$$

Whakareatia -\frac{3}{5} ki te \frac{3215}{11} 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.

$$x=-\frac{1929}{11}+90$$

Tāpiri 90 ki te -\frac{1929}{11}.

$$x=-\frac{939}{11}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{939}{11},y=\frac{3215}{11}$$

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.

$$5x+3y=450,3x+4y=913$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}450\\913\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&3\\3&4\end{matrix}\right))\left(\begin{matrix}5&3\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&4\end{matrix}\right))\left(\begin{matrix}450\\913\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&4\end{matrix}\right))\left(\begin{matrix}450\\913\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\3&4\end{matrix}\right))\left(\begin{matrix}450\\913\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-3\times 3}&-\frac{3}{5\times 4-3\times 3}\\-\frac{3}{5\times 4-3\times 3}&\frac{5}{5\times 4-3\times 3}\end{matrix}\right)\left(\begin{matrix}450\\913\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}&-\frac{3}{11}\\-\frac{3}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}450\\913\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\times 450-\frac{3}{11}\times 913\\-\frac{3}{11}\times 450+\frac{5}{11}\times 913\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{939}{11}\\\frac{3215}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{939}{11},y=\frac{3215}{11}$$

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.

$$5x+3y=450,3x+4y=913$$

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

$$3\times 5x+3\times 3y=3\times 450,5\times 3x+5\times 4y=5\times 913$$

Whakarūnātia.

$$15x+9y=1350,15x+20y=4565$$

Me tango 15x+20y=4565 mai i 15x+9y=1350 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+9y-20y=1350-4565$$

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.

$$9y-20y=1350-4565$$

Tāpiri 9y ki te -20y.

$$-11y=1350-4565$$

Tāpiri 1350 ki te -4565.

$$-11y=-3215$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{3215}{11}$$

Whakaurua te \frac{3215}{11} mō y ki 3x+4y=913. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\times \frac{3215}{11}=913$$

Whakareatia 4 ki te \frac{3215}{11}.

$$3x+\frac{12860}{11}=913$$

Me tango \frac{12860}{11} mai i ngā taha e rua o te whārite.

$$3x=-\frac{2817}{11}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{939}{11}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{939}{11},y=\frac{3215}{11}$$