Whakaoti i te {l}{7n+46=a}{11n+2=a}

$$\left\{ \begin{array} { l } { 7 n + 46 = a } \\ { 11 n + 2 = a } \end{array} \right.$$

Whakaoti mō n, a

$$n=11$$

$$a=123$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te a mai i ngā taha e rua.

$$7n+46-a=0$$

Tangohia te 46 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$7n-a=-46$$

Whakaarohia te whārite tuarua. Tangohia te a mai i ngā taha e rua.

$$11n+2-a=0$$

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$11n-a=-2$$

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.

$$7n-a=-46,11n-a=-2$$

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

$$7n-a=-46$$

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

$$7n=a-46$$

Whakawehea ngā taha e rua ki te 7.

$$n=\frac{1}{7}\left(a-46\right)$$

Whakareatia \frac{1}{7} ki te a-46.

$$n=\frac{1}{7}a-\frac{46}{7}$$

Whakakapia te \frac{-46+a}{7} mō te n ki tērā atu whārite, 11n-a=-2.

$$11\left(\frac{1}{7}a-\frac{46}{7}\right)-a=-2$$

Whakareatia 11 ki te \frac{-46+a}{7}.

$$\frac{11}{7}a-\frac{506}{7}-a=-2$$

Tāpiri \frac{11a}{7} ki te -a.

$$\frac{4}{7}a-\frac{506}{7}=-2$$

Me tāpiri \frac{506}{7} ki ngā taha e rua o te whārite.

$$\frac{4}{7}a=\frac{492}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$a=123$$

Whakaurua te 123 mō a ki n=\frac{1}{7}a-\frac{46}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō n hāngai tonu.

$$n=\frac{1}{7}\times 123-\frac{46}{7}$$

Whakareatia \frac{1}{7} ki te 123.

$$n=\frac{123-46}{7}$$

Tāpiri -\frac{46}{7} ki te \frac{123}{7} 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.

$$n=11$$

Kua oti te pūnaha te whakatau.

$$n=11,a=123$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te a mai i ngā taha e rua.

$$7n+46-a=0$$

Tangohia te 46 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$7n-a=-46$$

Whakaarohia te whārite tuarua. Tangohia te a mai i ngā taha e rua.

$$11n+2-a=0$$

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$11n-a=-2$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$7n-a=-46,11n-a=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right)\left(\begin{matrix}n\\a\end{matrix}\right)=\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-1\\11&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right)\left(\begin{matrix}n\\a\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}n\\a\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}n\\a\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}n\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7\left(-1\right)-\left(-11\right)}&-\frac{-1}{7\left(-1\right)-\left(-11\right)}\\-\frac{11}{7\left(-1\right)-\left(-11\right)}&\frac{7}{7\left(-1\right)-\left(-11\right)}\end{matrix}\right)\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}n\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\-\frac{11}{4}&\frac{7}{4}\end{matrix}\right)\left(\begin{matrix}-46\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}n\\a\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-46\right)+\frac{1}{4}\left(-2\right)\\-\frac{11}{4}\left(-46\right)+\frac{7}{4}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}n\\a\end{matrix}\right)=\left(\begin{matrix}11\\123\end{matrix}\right)$$

Tangohia ngā huānga poukapa n me a.

$$n=11,a=123$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te a mai i ngā taha e rua.

$$7n+46-a=0$$

Tangohia te 46 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$7n-a=-46$$

Whakaarohia te whārite tuarua. Tangohia te a mai i ngā taha e rua.

$$11n+2-a=0$$

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$11n-a=-2$$

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.

$$7n-a=-46,11n-a=-2$$

Me tango 11n-a=-2 mai i 7n-a=-46 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7n-11n-a+a=-46+2$$

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

$$7n-11n=-46+2$$

Tāpiri 7n ki te -11n.

$$-4n=-46+2$$

Tāpiri -46 ki te 2.

$$-4n=-44$$

Whakawehea ngā taha e rua ki te -4.

$$n=11$$

Whakaurua te 11 mō n ki 11n-a=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō a hāngai tonu.

$$11\times 11-a=-2$$

Whakareatia 11 ki te 11.

$$121-a=-2$$

Me tango 121 mai i ngā taha e rua o te whārite.

$$-a=-123$$

Whakawehea ngā taha e rua ki te -1.

$$a=123$$

Kua oti te pūnaha te whakatau.

$$n=11,a=123$$