Whakaoti i te {l}{-5x+5y=-10}{-2x+5y=-16}hline

$$\left. \begin{array} { l } { - 5 x + 5 y = - 10 } \\ { - 2 x + 5 y = - 16 } \\ \hline \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=-4$$

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.

$$-5x+5y=-10,-2x+5y=-16$$

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

$$-5x+5y=-10$$

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

$$-5x=-5y-10$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-5y-10\right)$$

Whakareatia -\frac{1}{5} ki te -5y-10.

$$x=y+2$$

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

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

Whakareatia -2 ki te y+2.

$$-2y-4+5y=-16$$

Tāpiri -2y ki te 5y.

$$3y-4=-16$$

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

$$x=-4+2$$

Tāpiri 2 ki te -4.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-4$$

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.

$$-5x+5y=-10,-2x+5y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right))\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right))\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right))\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&5\\-2&5\end{matrix}\right))\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{-5\times 5-5\left(-2\right)}&-\frac{5}{-5\times 5-5\left(-2\right)}\\-\frac{-2}{-5\times 5-5\left(-2\right)}&-\frac{5}{-5\times 5-5\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&\frac{1}{3}\\-\frac{2}{15}&\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}-10\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-10\right)+\frac{1}{3}\left(-16\right)\\-\frac{2}{15}\left(-10\right)+\frac{1}{3}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=-4$$

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.

$$-5x+5y=-10,-2x+5y=-16$$

Me tango -2x+5y=-16 mai i -5x+5y=-10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+2x+5y-5y=-10+16$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5x+2x=-10+16$$

Tāpiri -5x ki te 2x.

$$-3x=-10+16$$

Tāpiri -10 ki te 16.

$$-3x=6$$

Whakawehea ngā taha e rua ki te -3.

$$x=-2$$

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

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

Whakareatia -2 ki te -2.

$$4+5y=-16$$

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

$$5y=-20$$

Whakawehea ngā taha e rua ki te 5.

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=-4$$