Whakaoti i te {l}{-3x+5y=-16}{-5x-4y=-2}

$$\left\{ \begin{array} { l } { - 3 x + 5 y = - 16 } \\ { - 5 x - 4 y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=-2$$

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.

$$-3x+5y=-16,-5x-4y=-2$$

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

$$-3x+5y=-16$$

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

$$-3x=-5y-16$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(-5y-16\right)$$

Whakareatia -\frac{1}{3} ki te -5y-16.

$$x=\frac{5}{3}y+\frac{16}{3}$$

Whakakapia te \frac{5y+16}{3} mō te x ki tērā atu whārite, -5x-4y=-2.

$$-5\left(\frac{5}{3}y+\frac{16}{3}\right)-4y=-2$$

Whakareatia -5 ki te \frac{5y+16}{3}.

$$-\frac{25}{3}y-\frac{80}{3}-4y=-2$$

Tāpiri -\frac{25y}{3} ki te -4y.

$$-\frac{37}{3}y-\frac{80}{3}=-2$$

Me tāpiri \frac{80}{3} ki ngā taha e rua o te whārite.

$$-\frac{37}{3}y=\frac{74}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{37}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=\frac{5}{3}y+\frac{16}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{3}\left(-2\right)+\frac{16}{3}$$

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

$$x=\frac{-10+16}{3}$$

Tāpiri \frac{16}{3} ki te -\frac{10}{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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-2$$

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.

$$-3x+5y=-16,-5x-4y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right))\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right))\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right))\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&5\\-5&-4\end{matrix}\right))\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-3\left(-4\right)-5\left(-5\right)}&-\frac{5}{-3\left(-4\right)-5\left(-5\right)}\\-\frac{-5}{-3\left(-4\right)-5\left(-5\right)}&-\frac{3}{-3\left(-4\right)-5\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{37}&-\frac{5}{37}\\\frac{5}{37}&-\frac{3}{37}\end{matrix}\right)\left(\begin{matrix}-16\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{37}\left(-16\right)-\frac{5}{37}\left(-2\right)\\\frac{5}{37}\left(-16\right)-\frac{3}{37}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=-2$$

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.

$$-3x+5y=-16,-5x-4y=-2$$

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.

$$-5\left(-3\right)x-5\times 5y=-5\left(-16\right),-3\left(-5\right)x-3\left(-4\right)y=-3\left(-2\right)$$

Whakarūnātia.

$$15x-25y=80,15x+12y=6$$

Me tango 15x+12y=6 mai i 15x-25y=80 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-25y-12y=80-6$$

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.

$$-25y-12y=80-6$$

Tāpiri -25y ki te -12y.

$$-37y=80-6$$

Tāpiri 80 ki te -6.

$$-37y=74$$

Whakawehea ngā taha e rua ki te -37.

$$y=-2$$

Whakaurua te -2 mō y ki -5x-4y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x-4\left(-2\right)=-2$$

Whakareatia -4 ki te -2.

$$-5x+8=-2$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-5x=-10$$

Whakawehea ngā taha e rua ki te -5.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=-2$$