Whakaoti i te {l}{5u+x=-10}{3u+3x=0}

$$\left. \begin{array} { l } { 5 u + x = - 10 } \\ { 3 u + 3 x = 0 } \end{array} \right.$$

Whakaoti mō u, x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$u = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

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.

$$5u+x=-10,3u+3x=0$$

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

$$5u+x=-10$$

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

$$5u=-x-10$$

Whakawehea ngā taha e rua ki te 5.

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

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

$$u=-\frac{1}{5}x-2$$

Whakakapia te -\frac{x}{5}-2 mō te u ki tērā atu whārite, 3u+3x=0.

$$3\left(-\frac{1}{5}x-2\right)+3x=0$$

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

$$-\frac{3}{5}x-6+3x=0$$

Tāpiri -\frac{3x}{5} ki te 3x.

$$\frac{12}{5}x-6=0$$

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

$$\frac{12}{5}x=6$$

Whakawehea ngā taha e rua o te whārite ki te \frac{12}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{5}{2}$$

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

$$u=-\frac{1}{5}\times \frac{5}{2}-2$$

Whakareatia -\frac{1}{5} ki te \frac{5}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$u=-\frac{1}{2}-2$$

Tāpiri -2 ki te -\frac{1}{2}.

$$u=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$u=-\frac{5}{2},x=\frac{5}{2}$$

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.

$$5u+x=-10,3u+3x=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\3&3\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\3&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&1\\3&3\end{matrix}\right))\left(\begin{matrix}5&1\\3&3\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\3&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\3&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}u\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\3&3\end{matrix}\right))\left(\begin{matrix}-10\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-3}&-\frac{1}{5\times 3-3}\\-\frac{3}{5\times 3-3}&\frac{5}{5\times 3-3}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}&-\frac{1}{12}\\-\frac{1}{4}&\frac{5}{12}\end{matrix}\right)\left(\begin{matrix}-10\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\left(-10\right)\\-\frac{1}{4}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}u\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\\frac{5}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa u me x.

$$u=-\frac{5}{2},x=\frac{5}{2}$$

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.

$$5u+x=-10,3u+3x=0$$

Kia ōrite ai a 5u me 3u, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$3\times 5u+3x=3\left(-10\right),5\times 3u+5\times 3x=0$$

Whakarūnātia.

$$15u+3x=-30,15u+15x=0$$

Me tango 15u+15x=0 mai i 15u+3x=-30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15u-15u+3x-15x=-30$$

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

$$3x-15x=-30$$

Tāpiri 3x ki te -15x.

$$-12x=-30$$

Whakawehea ngā taha e rua ki te -12.

$$x=\frac{5}{2}$$

Whakaurua te \frac{5}{2} mō x ki 3u+3x=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō u hāngai tonu.

$$3u+3\times \frac{5}{2}=0$$

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

$$3u+\frac{15}{2}=0$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$3u=-\frac{15}{2}$$

Whakawehea ngā taha e rua ki te 3.

$$u=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$u=-\frac{5}{2},x=\frac{5}{2}$$