Whakaoti i te {l}{-3y+4x=13}{-5y-6x=-67}

$$\left. \begin{array} { l } { - 3 y + 4 x = 13 } \\ { - 5 y - 6 x = - 67 } \end{array} \right.$$

Whakaoti mō y, x

$$x=7$$

$$y=5$$

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.

$$-3y+4x=13,-5y-6x=-67$$

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

$$-3y+4x=13$$

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

$$-3y=-4x+13$$

Whakawehea ngā taha e rua ki te -3.

$$y=-\frac{1}{3}\left(-4x+13\right)$$

Whakareatia -\frac{1}{3} ki te -4x+13.

$$y=\frac{4}{3}x-\frac{13}{3}$$

Whakakapia te \frac{4x-13}{3} mō te y ki tērā atu whārite, -5y-6x=-67.

$$-5\left(\frac{4}{3}x-\frac{13}{3}\right)-6x=-67$$

Whakareatia -5 ki te \frac{4x-13}{3}.

$$-\frac{20}{3}x+\frac{65}{3}-6x=-67$$

Tāpiri -\frac{20x}{3} ki te -6x.

$$-\frac{38}{3}x+\frac{65}{3}=-67$$

Me tango \frac{65}{3} mai i ngā taha e rua o te whārite.

$$-\frac{38}{3}x=-\frac{266}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{38}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=7$$

Whakaurua te 7 mō x ki y=\frac{4}{3}x-\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{4}{3}\times 7-\frac{13}{3}$$

Whakareatia \frac{4}{3} ki te 7.

$$y=\frac{28-13}{3}$$

Tāpiri -\frac{13}{3} ki te \frac{28}{3} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=7$$

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.

$$-3y+4x=13,-5y-6x=-67$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}13\\-67\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right))\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right))\left(\begin{matrix}13\\-67\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right))\left(\begin{matrix}13\\-67\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}-3&4\\-5&-6\end{matrix}\right))\left(\begin{matrix}13\\-67\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{-3\left(-6\right)-4\left(-5\right)}&-\frac{4}{-3\left(-6\right)-4\left(-5\right)}\\-\frac{-5}{-3\left(-6\right)-4\left(-5\right)}&-\frac{3}{-3\left(-6\right)-4\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}13\\-67\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}&-\frac{2}{19}\\\frac{5}{38}&-\frac{3}{38}\end{matrix}\right)\left(\begin{matrix}13\\-67\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}\times 13-\frac{2}{19}\left(-67\right)\\\frac{5}{38}\times 13-\frac{3}{38}\left(-67\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=5,x=7$$

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.

$$-3y+4x=13,-5y-6x=-67$$

Kia ōrite ai a -3y me -5y, 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 -3.

$$-5\left(-3\right)y-5\times 4x=-5\times 13,-3\left(-5\right)y-3\left(-6\right)x=-3\left(-67\right)$$

Whakarūnātia.

$$15y-20x=-65,15y+18x=201$$

Me tango 15y+18x=201 mai i 15y-20x=-65 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15y-15y-20x-18x=-65-201$$

Tāpiri 15y ki te -15y. Ka whakakore atu ngā kupu 15y me -15y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20x-18x=-65-201$$

Tāpiri -20x ki te -18x.

$$-38x=-65-201$$

Tāpiri -65 ki te -201.

$$-38x=-266$$

Whakawehea ngā taha e rua ki te -38.

$$x=7$$

Whakaurua te 7 mō x ki -5y-6x=-67. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-5y-6\times 7=-67$$

Whakareatia -6 ki te 7.

$$-5y-42=-67$$

Me tāpiri 42 ki ngā taha e rua o te whārite.

$$-5y=-25$$

Whakawehea ngā taha e rua ki te -5.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$y=5,x=7$$