Whakaoti i te {l}{5y-4z=-1}{-7y+7z=9}

$$\left\{ \begin{array} { l } { 5 y - 4 z = - 1 } \\ { - 7 y + 7 z = 9 } \end{array} \right.$$

Whakaoti mō y, z

$$y = \frac{29}{7} = 4\frac{1}{7} \approx 4.142857143$$

$$z = \frac{38}{7} = 5\frac{3}{7} \approx 5.428571429$$

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.

$$5y-4z=-1,-7y+7z=9$$

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

$$5y-4z=-1$$

Me tāpiri 4z ki ngā taha e rua o te whārite.

$$5y=4z-1$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{1}{5}\left(4z-1\right)$$

Whakareatia \frac{1}{5} ki te 4z-1.

$$y=\frac{4}{5}z-\frac{1}{5}$$

Whakakapia te \frac{4z-1}{5} mō te y ki tērā atu whārite, -7y+7z=9.

$$-7\left(\frac{4}{5}z-\frac{1}{5}\right)+7z=9$$

Whakareatia -7 ki te \frac{4z-1}{5}.

$$-\frac{28}{5}z+\frac{7}{5}+7z=9$$

Tāpiri -\frac{28z}{5} ki te 7z.

$$\frac{7}{5}z+\frac{7}{5}=9$$

Me tango \frac{7}{5} mai i ngā taha e rua o te whārite.

$$\frac{7}{5}z=\frac{38}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=\frac{38}{7}$$

Whakaurua te \frac{38}{7} mō z ki y=\frac{4}{5}z-\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{4}{5}\times \frac{38}{7}-\frac{1}{5}$$

Whakareatia \frac{4}{5} ki te \frac{38}{7} 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.

$$y=\frac{152}{35}-\frac{1}{5}$$

Tāpiri -\frac{1}{5} ki te \frac{152}{35} 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.

$$y=\frac{29}{7}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{29}{7},z=\frac{38}{7}$$

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.

$$5y-4z=-1,-7y+7z=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\-7&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right))\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-7&7\end{matrix}\right))\left(\begin{matrix}-1\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-\left(-4\left(-7\right)\right)}&-\frac{-4}{5\times 7-\left(-4\left(-7\right)\right)}\\-\frac{-7}{5\times 7-\left(-4\left(-7\right)\right)}&\frac{5}{5\times 7-\left(-4\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}1&\frac{4}{7}\\1&\frac{5}{7}\end{matrix}\right)\left(\begin{matrix}-1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-1+\frac{4}{7}\times 9\\-1+\frac{5}{7}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}\frac{29}{7}\\\frac{38}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me z.

$$y=\frac{29}{7},z=\frac{38}{7}$$

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.

$$5y-4z=-1,-7y+7z=9$$

Kia ōrite ai a 5y me -7y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-7\times 5y-7\left(-4\right)z=-7\left(-1\right),5\left(-7\right)y+5\times 7z=5\times 9$$

Whakarūnātia.

$$-35y+28z=7,-35y+35z=45$$

Me tango -35y+35z=45 mai i -35y+28z=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-35y+35y+28z-35z=7-45$$

Tāpiri -35y ki te 35y. Ka whakakore atu ngā kupu -35y me 35y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$28z-35z=7-45$$

Tāpiri 28z ki te -35z.

$$-7z=7-45$$

Tāpiri 7 ki te -45.

$$-7z=-38$$

Whakawehea ngā taha e rua ki te -7.

$$z=\frac{38}{7}$$

Whakaurua te \frac{38}{7} mō z ki -7y+7z=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-7y+7\times \frac{38}{7}=9$$

Whakareatia 7 ki te \frac{38}{7}.

$$-7y+38=9$$

Me tango 38 mai i ngā taha e rua o te whārite.

$$-7y=-29$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{29}{7}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{29}{7},z=\frac{38}{7}$$