Whakaoti i te {l}{-x-5y=14}{-2x-7y=16}

$$\left. \begin{array} { l } { - x - 5 y = 14 } \\ { - 2 x - 7 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-4$$

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.

$$-x-5y=14,-2x-7y=16$$

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

$$-x-5y=14$$

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

$$-x=5y+14$$

Whakawehea ngā taha e rua ki te -1.

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

Whakareatia -1 ki te 5y+14.

$$x=-5y-14$$

Whakakapia te -5y-14 mō te x ki tērā atu whārite, -2x-7y=16.

$$-2\left(-5y-14\right)-7y=16$$

Whakareatia -2 ki te -5y-14.

$$10y+28-7y=16$$

Tāpiri 10y ki te -7y.

$$3y+28=16$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$3y=-12$$

Whakawehea ngā taha e rua ki te 3.

$$y=-4$$

Whakaurua te -4 mō y ki x=-5y-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\left(-4\right)-14$$

Whakareatia -5 ki te -4.

$$x=20-14$$

Tāpiri -14 ki te 20.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-4$$

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.

$$-x-5y=14,-2x-7y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right))\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&-5\\-2&-7\end{matrix}\right))\left(\begin{matrix}14\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-\left(-7\right)-\left(-5\left(-2\right)\right)}&-\frac{-5}{-\left(-7\right)-\left(-5\left(-2\right)\right)}\\-\frac{-2}{-\left(-7\right)-\left(-5\left(-2\right)\right)}&-\frac{1}{-\left(-7\right)-\left(-5\left(-2\right)\right)}\end{matrix}\right)\left(\begin{matrix}14\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}&-\frac{5}{3}\\-\frac{2}{3}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}14\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}\times 14-\frac{5}{3}\times 16\\-\frac{2}{3}\times 14+\frac{1}{3}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-4$$

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.

$$-x-5y=14,-2x-7y=16$$

Kia ōrite ai a -x 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 -1.

$$-2\left(-1\right)x-2\left(-5\right)y=-2\times 14,-\left(-2\right)x-\left(-7y\right)=-16$$

Whakarūnātia.

$$2x+10y=-28,2x+7y=-16$$

Me tango 2x+7y=-16 mai i 2x+10y=-28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2x-2x+10y-7y=-28+16$$

Tāpiri 2x ki te -2x. Ka whakakore atu ngā kupu 2x me -2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-7y=-28+16$$

Tāpiri 10y ki te -7y.

$$3y=-28+16$$

Tāpiri -28 ki te 16.

$$3y=-12$$

Whakawehea ngā taha e rua ki te 3.

$$y=-4$$

Whakaurua te -4 mō y ki -2x-7y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-7\left(-4\right)=16$$

Whakareatia -7 ki te -4.

$$-2x+28=16$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-2x=-12$$

Whakawehea ngā taha e rua ki te -2.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-4$$