Whakaoti i te {l}{-3x-5y=17}{-5x+6y=14}

$$\left. \begin{array} { l } { - 3 x - 5 y = 17 } \\ { - 5 x + 6 y = 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-4$$

$$y=-1$$

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.

$$-3x-5y=17,-5x+6y=14$$

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

$$-3x-5y=17$$

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

$$-3x=5y+17$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(5y+17\right)$$

Whakareatia -\frac{1}{3} ki te 5y+17.

$$x=-\frac{5}{3}y-\frac{17}{3}$$

Whakakapia te \frac{-5y-17}{3} mō te x ki tērā atu whārite, -5x+6y=14.

$$-5\left(-\frac{5}{3}y-\frac{17}{3}\right)+6y=14$$

Whakareatia -5 ki te \frac{-5y-17}{3}.

$$\frac{25}{3}y+\frac{85}{3}+6y=14$$

Tāpiri \frac{25y}{3} ki te 6y.

$$\frac{43}{3}y+\frac{85}{3}=14$$

Me tango \frac{85}{3} mai i ngā taha e rua o te whārite.

$$\frac{43}{3}y=-\frac{43}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{43}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{5}{3}y-\frac{17}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\left(-1\right)-\frac{17}{3}$$

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

$$x=\frac{5-17}{3}$$

Tāpiri -\frac{17}{3} ki te \frac{5}{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.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-1$$

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.

$$-3x-5y=17,-5x+6y=14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}17\\14\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right))\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right))\left(\begin{matrix}17\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right))\left(\begin{matrix}17\\14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&-5\\-5&6\end{matrix}\right))\left(\begin{matrix}17\\14\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{-3\times 6-\left(-5\left(-5\right)\right)}&-\frac{-5}{-3\times 6-\left(-5\left(-5\right)\right)}\\-\frac{-5}{-3\times 6-\left(-5\left(-5\right)\right)}&-\frac{3}{-3\times 6-\left(-5\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}17\\14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{43}&-\frac{5}{43}\\-\frac{5}{43}&\frac{3}{43}\end{matrix}\right)\left(\begin{matrix}17\\14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{43}\times 17-\frac{5}{43}\times 14\\-\frac{5}{43}\times 17+\frac{3}{43}\times 14\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-4,y=-1$$

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.

$$-3x-5y=17,-5x+6y=14$$

Kia ōrite ai a -3x 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 -3.

$$-5\left(-3\right)x-5\left(-5\right)y=-5\times 17,-3\left(-5\right)x-3\times 6y=-3\times 14$$

Whakarūnātia.

$$15x+25y=-85,15x-18y=-42$$

Me tango 15x-18y=-42 mai i 15x+25y=-85 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+25y+18y=-85+42$$

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

$$25y+18y=-85+42$$

Tāpiri 25y ki te 18y.

$$43y=-85+42$$

Tāpiri -85 ki te 42.

$$43y=-43$$

Whakawehea ngā taha e rua ki te 43.

$$y=-1$$

Whakaurua te -1 mō y ki -5x+6y=14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+6\left(-1\right)=14$$

Whakareatia 6 ki te -1.

$$-5x-6=14$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-5x=20$$

Whakawehea ngā taha e rua ki te -5.

$$x=-4$$

Kua oti te pūnaha te whakatau.

$$x=-4,y=-1$$