Whakaoti i te {c}{x-3=y}{4x=37+3y}

$$\left. \begin{array} { c } { x - 3 = y } \\ { 4 x = 37 + 3 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=28$$

$$y=25$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$4x-3y=37$$

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.

$$x-y=3,4x-3y=37$$

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

$$x-y=3$$

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

$$x=y+3$$

Whakakapia te y+3 mō te x ki tērā atu whārite, 4x-3y=37.

$$4\left(y+3\right)-3y=37$$

Whakareatia 4 ki te y+3.

$$4y+12-3y=37$$

Tāpiri 4y ki te -3y.

$$y+12=37$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$y=25$$

Whakaurua te 25 mō y ki x=y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=25+3$$

Tāpiri 3 ki te 25.

$$x=28$$

Kua oti te pūnaha te whakatau.

$$x=28,y=25$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$4x-3y=37$$

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

$$x-y=3,4x-3y=37$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\37\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-1\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right))\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right))\left(\begin{matrix}3\\37\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right))\left(\begin{matrix}3\\37\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\4&-3\end{matrix}\right))\left(\begin{matrix}3\\37\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-\left(-4\right)}&-\frac{-1}{-3-\left(-4\right)}\\-\frac{4}{-3-\left(-4\right)}&\frac{1}{-3-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}3\\37\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&1\\-4&1\end{matrix}\right)\left(\begin{matrix}3\\37\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\times 3+37\\-4\times 3+37\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\25\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=28,y=25$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$4x-3y=37$$

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.

$$x-y=3,4x-3y=37$$

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4x+4\left(-1\right)y=4\times 3,4x-3y=37$$

Whakarūnātia.

$$4x-4y=12,4x-3y=37$$

Me tango 4x-3y=37 mai i 4x-4y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x-4y+3y=12-37$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y+3y=12-37$$

Tāpiri -4y ki te 3y.

$$-y=12-37$$

Tāpiri 12 ki te -37.

$$-y=-25$$

Whakawehea ngā taha e rua ki te -1.

$$y=25$$

Whakaurua te 25 mō y ki 4x-3y=37. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\times 25=37$$

Whakareatia -3 ki te 25.

$$4x-75=37$$

Me tāpiri 75 ki ngā taha e rua o te whārite.

$$4x=112$$

Whakawehea ngā taha e rua ki te 4.

$$x=28$$

Kua oti te pūnaha te whakatau.

$$x=28,y=25$$