Whakaoti i te {l}{500y+150.25x=0}{2990y+225.75x=0}

$$\left. \begin{array} { l } { 500 y + 150.25 x = 0 } \\ { 2990 y + 225.75 x = 0 } \end{array} \right.$$

Whakaoti mō y, x

$$x=0$$

$$y=0$$

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.

$$500y+150.25x=0,2990y+225.75x=0$$

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

$$500y+150.25x=0$$

Me tango \frac{601x}{4} mai i ngā taha e rua o te whārite.

$$500y=-150.25x$$

Whakawehea ngā taha e rua ki te 500.

$$y=\frac{1}{500}\left(-150.25\right)x$$

Whakareatia \frac{1}{500} ki te -\frac{601x}{4}.

$$y=-\frac{601}{2000}x$$

Whakakapia te -\frac{601x}{2000} mō te y ki tērā atu whārite, 2990y+225.75x=0.

$$2990\left(-\frac{601}{2000}\right)x+225.75x=0$$

Whakareatia 2990 ki te -\frac{601x}{2000}.

$$-\frac{179699}{200}x+225.75x=0$$

Tāpiri -\frac{179699x}{200} ki te \frac{903x}{4}.

$$-\frac{134549}{200}x=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{134549}{200}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0$$

Whakaurua te 0 mō x ki y=-\frac{601}{2000}x. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$y=0,x=0$$

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.

$$500y+150.25x=0,2990y+225.75x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right).

$$inverse(\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right))\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}500&150.25\\2990&225.75\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{225.75}{500\times 225.75-150.25\times 2990}&-\frac{150.25}{500\times 225.75-150.25\times 2990}\\-\frac{2990}{500\times 225.75-150.25\times 2990}&\frac{500}{500\times 225.75-150.25\times 2990}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{903}{1345490}&\frac{601}{1345490}\\\frac{1196}{134549}&-\frac{200}{134549}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=0,x=0$$

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.

$$500y+150.25x=0,2990y+225.75x=0$$

Kia ōrite ai a 500y me 2990y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2990 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 500.

$$2990\times 500y+2990\times 150.25x=0,500\times 2990y+500\times 225.75x=0$$

Whakarūnātia.

$$1495000y+449247.5x=0,1495000y+112875x=0$$

Me tango 1495000y+112875x=0 mai i 1495000y+449247.5x=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$1495000y-1495000y+449247.5x-112875x=0$$

Tāpiri 1495000y ki te -1495000y. Ka whakakore atu ngā kupu 1495000y me -1495000y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$449247.5x-112875x=0$$

Tāpiri \frac{898495x}{2} ki te -112875x.

$$336372.5x=0$$

Whakawehea ngā taha e rua o te whārite ki te 336372.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=0$$

Whakaurua te 0 mō x ki 2990y+225.75x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2990y=0$$

Whakawehea ngā taha e rua ki te 2990.

$$y=0$$

Kua oti te pūnaha te whakatau.

$$y=0,x=0$$