Whakaoti i te {l}{3c+5z=-15}{5c+3z=-9}

$$\left. \begin{array} { l } { 3 c + 5 z = - 15 } \\ { 5 c + 3 z = - 9 } \end{array} \right.$$

Whakaoti mō c, z

$$z=-3$$

$$c=0$$

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.

$$3c+5z=-15,5c+3z=-9$$

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

$$3c+5z=-15$$

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

$$3c=-5z-15$$

Whakawehea ngā taha e rua ki te 3.

$$c=\frac{1}{3}\left(-5z-15\right)$$

Whakareatia \frac{1}{3} ki te -5z-15.

$$c=-\frac{5}{3}z-5$$

Whakakapia te -\frac{5z}{3}-5 mō te c ki tērā atu whārite, 5c+3z=-9.

$$5\left(-\frac{5}{3}z-5\right)+3z=-9$$

Whakareatia 5 ki te -\frac{5z}{3}-5.

$$-\frac{25}{3}z-25+3z=-9$$

Tāpiri -\frac{25z}{3} ki te 3z.

$$-\frac{16}{3}z-25=-9$$

Me tāpiri 25 ki ngā taha e rua o te whārite.

$$-\frac{16}{3}z=16$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{16}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=-3$$

Whakaurua te -3 mō z ki c=-\frac{5}{3}z-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$c=-\frac{5}{3}\left(-3\right)-5$$

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

$$c=5-5$$

Tāpiri -5 ki te 5.

$$c=0$$

Kua oti te pūnaha te whakatau.

$$c=0,z=-3$$

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.

$$3c+5z=-15,5c+3z=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\5&3\end{matrix}\right)\left(\begin{matrix}c\\z\end{matrix}\right)=\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&5\\5&3\end{matrix}\right))\left(\begin{matrix}3&5\\5&3\end{matrix}\right)\left(\begin{matrix}c\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&3\end{matrix}\right))\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&3\end{matrix}\right))\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\z\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&3\end{matrix}\right))\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\z\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-5\times 5}&-\frac{5}{3\times 3-5\times 5}\\-\frac{5}{3\times 3-5\times 5}&\frac{3}{3\times 3-5\times 5}\end{matrix}\right)\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}&\frac{5}{16}\\\frac{5}{16}&-\frac{3}{16}\end{matrix}\right)\left(\begin{matrix}-15\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}c\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{16}\left(-15\right)+\frac{5}{16}\left(-9\right)\\\frac{5}{16}\left(-15\right)-\frac{3}{16}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}c\\z\end{matrix}\right)=\left(\begin{matrix}0\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa c me z.

$$c=0,z=-3$$

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.

$$3c+5z=-15,5c+3z=-9$$

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

$$5\times 3c+5\times 5z=5\left(-15\right),3\times 5c+3\times 3z=3\left(-9\right)$$

Whakarūnātia.

$$15c+25z=-75,15c+9z=-27$$

Me tango 15c+9z=-27 mai i 15c+25z=-75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15c-15c+25z-9z=-75+27$$

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

$$25z-9z=-75+27$$

Tāpiri 25z ki te -9z.

$$16z=-75+27$$

Tāpiri -75 ki te 27.

$$16z=-48$$

Whakawehea ngā taha e rua ki te 16.

$$z=-3$$

Whakaurua te -3 mō z ki 5c+3z=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō c hāngai tonu.

$$5c+3\left(-3\right)=-9$$

Whakareatia 3 ki te -3.

$$5c-9=-9$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$5c=0$$

Whakawehea ngā taha e rua ki te 5.

$$c=0$$

Kua oti te pūnaha te whakatau.

$$c=0,z=-3$$