Whakaoti i te {l}{2x-y=-2}{y=5x-1}

$$\left. \begin{array} { l } { 2 x - y = - 2 } \\ { y = 5 x - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=-1$$

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.

$$2x-y=-2,-5x+y=-1$$

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

$$2x-y=-2$$

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

$$2x=y-2$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y-2\right)$$

Whakareatia \frac{1}{2} ki te y-2.

$$x=\frac{1}{2}y-1$$

Whakakapia te \frac{y}{2}-1 mō te x ki tērā atu whārite, -5x+y=-1.

$$-5\left(\frac{1}{2}y-1\right)+y=-1$$

Whakareatia -5 ki te \frac{y}{2}-1.

$$-\frac{5}{2}y+5+y=-1$$

Tāpiri -\frac{5y}{2} ki te y.

$$-\frac{3}{2}y+5=-1$$

Me tango 5 mai i ngā taha e rua o te whārite.

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

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

$$y=4$$

Whakaurua te 4 mō y ki x=\frac{1}{2}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times 4-1$$

Whakareatia \frac{1}{2} ki te 4.

$$x=2-1$$

Tāpiri -1 ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=-1$$

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

$$2x-y=-2,-5x+y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\-5&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right))\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right))\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right))\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\-5&1\end{matrix}\right))\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-\left(-\left(-5\right)\right)}&-\frac{-1}{2-\left(-\left(-5\right)\right)}\\-\frac{-5}{2-\left(-\left(-5\right)\right)}&\frac{2}{2-\left(-\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&-\frac{1}{3}\\-\frac{5}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}-2\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-2\right)-\frac{1}{3}\left(-1\right)\\-\frac{5}{3}\left(-2\right)-\frac{2}{3}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=-1$$

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.

$$2x-y=-2,-5x+y=-1$$

Kia ōrite ai a 2x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$-5\times 2x-5\left(-1\right)y=-5\left(-2\right),2\left(-5\right)x+2y=2\left(-1\right)$$

Whakarūnātia.

$$-10x+5y=10,-10x+2y=-2$$

Me tango -10x+2y=-2 mai i -10x+5y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-10x+10x+5y-2y=10+2$$

Tāpiri -10x ki te 10x. Ka whakakore atu ngā kupu -10x me 10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-2y=10+2$$

Tāpiri 5y ki te -2y.

$$3y=10+2$$

Tāpiri 10 ki te 2.

$$3y=12$$

Whakawehea ngā taha e rua ki te 3.

$$y=4$$

Whakaurua te 4 mō y ki -5x+y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+4=-1$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=4$$