Whakaoti i te {l}{2+3x-y=20}{x-2=y+2}

$$\left. \begin{array} { l } { 2 + 3 x - y = 20 } \\ { x - 2 = y + 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2 mai i ngā taha e rua.

$$3x-y=20-2$$

Tangohia te 2 i te 20, ka 18.

$$3x-y=18$$

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

$$x-2-y=2$$

Me tāpiri te 2 ki ngā taha e rua.

$$x-y=2+2$$

Tāpirihia te 2 ki te 2, ka 4.

$$x-y=4$$

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.

$$3x-y=18,x-y=4$$

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

$$3x-y=18$$

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

$$3x=y+18$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+18\right)$$

Whakareatia \frac{1}{3} ki te y+18.

$$x=\frac{1}{3}y+6$$

Whakakapia te \frac{y}{3}+6 mō te x ki tērā atu whārite, x-y=4.

$$\frac{1}{3}y+6-y=4$$

Tāpiri \frac{y}{3} ki te -y.

$$-\frac{2}{3}y+6=4$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-\frac{2}{3}y=-2$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{1}{3}y+6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 3+6$$

Whakareatia \frac{1}{3} ki te 3.

$$x=1+6$$

Tāpiri 6 ki te 1.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2 mai i ngā taha e rua.

$$3x-y=20-2$$

Tangohia te 2 i te 20, ka 18.

$$3x-y=18$$

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

$$x-2-y=2$$

Me tāpiri te 2 ki ngā taha e rua.

$$x-y=2+2$$

Tāpirihia te 2 ki te 2, ka 4.

$$x-y=4$$

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

$$3x-y=18,x-y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right))\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right))\left(\begin{matrix}18\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right))\left(\begin{matrix}18\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-1\end{matrix}\right))\left(\begin{matrix}18\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-1\right)}&-\frac{-1}{3\left(-1\right)-\left(-1\right)}\\-\frac{1}{3\left(-1\right)-\left(-1\right)}&\frac{3}{3\left(-1\right)-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}18\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{1}{2}&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}18\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 18-\frac{1}{2}\times 4\\\frac{1}{2}\times 18-\frac{3}{2}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2 mai i ngā taha e rua.

$$3x-y=20-2$$

Tangohia te 2 i te 20, ka 18.

$$3x-y=18$$

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

$$x-2-y=2$$

Me tāpiri te 2 ki ngā taha e rua.

$$x-y=2+2$$

Tāpirihia te 2 ki te 2, ka 4.

$$x-y=4$$

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.

$$3x-y=18,x-y=4$$

Me tango x-y=4 mai i 3x-y=18 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-x-y+y=18-4$$

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

$$3x-x=18-4$$

Tāpiri 3x ki te -x.

$$2x=18-4$$

Tāpiri 18 ki te -4.

$$2x=14$$

Whakawehea ngā taha e rua ki te 2.

$$x=7$$

Whakaurua te 7 mō x ki x-y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$7-y=4$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$-y=-3$$

Kua oti te pūnaha te whakatau.

$$x=7,y=3$$