Whakaoti i te {l}{x+y=90}{x+36=3y}

$$\left. \begin{array} { l } { x + y = 90 } \\ { x + 36 = 3 y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{117}{2} = 58\frac{1}{2} = 58.5$$

$$y = \frac{63}{2} = 31\frac{1}{2} = 31.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+36-3y=0$$

Tangohia te 36 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-36$$

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.

$$x+y=90,x-3y=-36$$

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, x-3y=-36.

$$-y+90-3y=-36$$

Tāpiri -y ki te -3y.

$$-4y+90=-36$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$-4y=-126$$

Whakawehea ngā taha e rua ki te -4.

$$y=\frac{63}{2}$$

Whakaurua te \frac{63}{2} 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.

$$x=-\frac{63}{2}+90$$

Tāpiri 90 ki te -\frac{63}{2}.

$$x=\frac{117}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{117}{2},y=\frac{63}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+36-3y=0$$

Tangohia te 36 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-36$$

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

$$x+y=90,x-3y=-36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}90\\-36\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}1&1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}90\\-36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}90\\-36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-3\end{matrix}\right))\left(\begin{matrix}90\\-36\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-1}&-\frac{1}{-3-1}\\-\frac{1}{-3-1}&\frac{1}{-3-1}\end{matrix}\right)\left(\begin{matrix}90\\-36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}&\frac{1}{4}\\\frac{1}{4}&-\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}90\\-36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4}\times 90+\frac{1}{4}\left(-36\right)\\\frac{1}{4}\times 90-\frac{1}{4}\left(-36\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{117}{2}\\\frac{63}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{117}{2},y=\frac{63}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 3y mai i ngā taha e rua.

$$x+36-3y=0$$

Tangohia te 36 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x-3y=-36$$

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.

$$x+y=90,x-3y=-36$$

Me tango x-3y=-36 mai i x+y=90 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+3y=90+36$$

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

$$y+3y=90+36$$

Tāpiri y ki te 3y.

$$4y=90+36$$

Tāpiri 90 ki te 36.

$$4y=126$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{63}{2}$$

Whakaurua te \frac{63}{2} mō y ki x-3y=-36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times \frac{63}{2}=-36$$

Whakareatia -3 ki te \frac{63}{2}.

$$x-\frac{189}{2}=-36$$

Me tāpiri \frac{189}{2} ki ngā taha e rua o te whārite.

$$x=\frac{117}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{117}{2},y=\frac{63}{2}$$