Whakaoti i te {l}{3x-5y=-6}{2x-3y=-5}

$$\left. \begin{array} { l } { 3 x - 5 y = - 6 } \\ { 2 x - 3 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=-3$$

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.

$$3x-5y=-6,2x-3y=-5$$

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

$$3x-5y=-6$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$3x=5y-6$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(5y-6\right)$$

Whakareatia \frac{1}{3} ki te 5y-6.

$$x=\frac{5}{3}y-2$$

Whakakapia te \frac{5y}{3}-2 mō te x ki tērā atu whārite, 2x-3y=-5.

$$2\left(\frac{5}{3}y-2\right)-3y=-5$$

Whakareatia 2 ki te \frac{5y}{3}-2.

$$\frac{10}{3}y-4-3y=-5$$

Tāpiri \frac{10y}{3} ki te -3y.

$$\frac{1}{3}y-4=-5$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$\frac{1}{3}y=-1$$

Me whakarea ngā taha e rua ki te 3.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{5}{3}y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-3\right)-2$$

Whakareatia \frac{5}{3} ki te -3.

$$x=-5-2$$

Tāpiri -2 ki te -5.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-3$$

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.

$$3x-5y=-6,2x-3y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right))\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\2&-3\end{matrix}\right))\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-\left(-5\times 2\right)}&-\frac{-5}{3\left(-3\right)-\left(-5\times 2\right)}\\-\frac{2}{3\left(-3\right)-\left(-5\times 2\right)}&\frac{3}{3\left(-3\right)-\left(-5\times 2\right)}\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&5\\-2&3\end{matrix}\right)\left(\begin{matrix}-6\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\left(-6\right)+5\left(-5\right)\\-2\left(-6\right)+3\left(-5\right)\end{matrix}\right)$$

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

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.

$$3x-5y=-6,2x-3y=-5$$

Kia ōrite ai a 3x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$2\times 3x+2\left(-5\right)y=2\left(-6\right),3\times 2x+3\left(-3\right)y=3\left(-5\right)$$

Whakarūnātia.

$$6x-10y=-12,6x-9y=-15$$

Me tango 6x-9y=-15 mai i 6x-10y=-12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-10y+9y=-12+15$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y+9y=-12+15$$

Tāpiri -10y ki te 9y.

$$-y=-12+15$$

Tāpiri -12 ki te 15.

$$-y=3$$

Whakawehea ngā taha e rua ki te -1.

$$y=-3$$

Whakaurua te -3 mō y ki 2x-3y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-3\right)=-5$$

Whakareatia -3 ki te -3.

$$2x+9=-5$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$2x=-14$$

Whakawehea ngā taha e rua ki te 2.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=-3$$