Whakaoti i te {l}{-5x+10y=15}{-5x+2y=-1}

$$\left. \begin{array} { l } { - 5 x + 10 y = 15 } \\ { - 5 x + 2 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

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.

$$-5x+10y=15,-5x+2y=-1$$

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

$$-5x+10y=15$$

Me tango 10y mai i ngā taha e rua o te whārite.

$$-5x=-10y+15$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-10y+15\right)$$

Whakareatia -\frac{1}{5} ki te -10y+15.

$$x=2y-3$$

Whakakapia te 2y-3 mō te x ki tērā atu whārite, -5x+2y=-1.

$$-5\left(2y-3\right)+2y=-1$$

Whakareatia -5 ki te 2y-3.

$$-10y+15+2y=-1$$

Tāpiri -10y ki te 2y.

$$-8y+15=-1$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-8y=-16$$

Whakawehea ngā taha e rua ki te -8.

$$y=2$$

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

$$x=2\times 2-3$$

Whakareatia 2 ki te 2.

$$x=4-3$$

Tāpiri -3 ki te 4.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

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.

$$-5x+10y=15,-5x+2y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&10\\-5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right))\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right))\left(\begin{matrix}15\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right))\left(\begin{matrix}15\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&10\\-5&2\end{matrix}\right))\left(\begin{matrix}15\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{-5\times 2-10\left(-5\right)}&-\frac{10}{-5\times 2-10\left(-5\right)}\\-\frac{-5}{-5\times 2-10\left(-5\right)}&-\frac{5}{-5\times 2-10\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}15\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}&-\frac{1}{4}\\\frac{1}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}15\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}\times 15-\frac{1}{4}\left(-1\right)\\\frac{1}{8}\times 15-\frac{1}{8}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=2$$

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.

$$-5x+10y=15,-5x+2y=-1$$

Me tango -5x+2y=-1 mai i -5x+10y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x+10y-2y=15+1$$

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

$$10y-2y=15+1$$

Tāpiri 10y ki te -2y.

$$8y=15+1$$

Tāpiri 15 ki te 1.

$$8y=16$$

Whakawehea ngā taha e rua ki te 8.

$$y=2$$

Whakaurua te 2 mō y ki -5x+2y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+2\times 2=-1$$

Whakareatia 2 ki te 2.

$$-5x+4=-1$$

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

$$-5x=-5$$

Whakawehea ngā taha e rua ki te -5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$