Whakaoti i te {l}{x+y=90}{3x-3y=90}

$$\left\{ \begin{array} { l } { x + y = 90 } \\ { 3 x - 3 y = 90 } \end{array} \right.$$

Whakaoti mō x, y

$$x=60$$

$$y=30$$

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.

$$x+y=90,3x-3y=90$$

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

$$x+y=90$$

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

$$x=-y+90$$

Whakakapia te -y+90 mō te x ki tērā atu whārite, 3x-3y=90.

$$3\left(-y+90\right)-3y=90$$

Whakareatia 3 ki te -y+90.

$$-3y+270-3y=90$$

Tāpiri -3y ki te -3y.

$$-6y+270=90$$

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

$$-6y=-180$$

Whakawehea ngā taha e rua ki te -6.

$$y=30$$

Whakaurua te 30 mō y ki x=-y+90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-30+90$$

Tāpiri 90 ki te -30.

$$x=60$$

Kua oti te pūnaha te whakatau.

$$x=60,y=30$$

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.

$$x+y=90,3x-3y=90$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}90\\90\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\3&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\3&-3\end{matrix}\right))\left(\begin{matrix}1&1\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&-3\end{matrix}\right))\left(\begin{matrix}90\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&-3\end{matrix}\right))\left(\begin{matrix}90\\90\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\3&-3\end{matrix}\right))\left(\begin{matrix}90\\90\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-3}&-\frac{1}{-3-3}\\-\frac{3}{-3-3}&\frac{1}{-3-3}\end{matrix}\right)\left(\begin{matrix}90\\90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{6}\\\frac{1}{2}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}90\\90\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 90+\frac{1}{6}\times 90\\\frac{1}{2}\times 90-\frac{1}{6}\times 90\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}60\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=60,y=30$$

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.

$$x+y=90,3x-3y=90$$

Kia ōrite ai a x 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 1.

$$3x+3y=3\times 90,3x-3y=90$$

Whakarūnātia.

$$3x+3y=270,3x-3y=90$$

Me tango 3x-3y=90 mai i 3x+3y=270 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+3y+3y=270-90$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+3y=270-90$$

Tāpiri 3y ki te 3y.

$$6y=270-90$$

Tāpiri 270 ki te -90.

$$6y=180$$

Whakawehea ngā taha e rua ki te 6.

$$y=30$$

Whakaurua te 30 mō y ki 3x-3y=90. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-3\times 30=90$$

Whakareatia -3 ki te 30.

$$3x-90=90$$

Me tāpiri 90 ki ngā taha e rua o te whārite.

$$3x=180$$

Whakawehea ngā taha e rua ki te 3.

$$x=60$$

Kua oti te pūnaha te whakatau.

$$x=60,y=30$$