Whakaoti i te {l}{7x-2y=30}{5x+2y=-156}

$$\left\{ \begin{array} { l } { 7 x - 2 y = 30 } \\ { 5 x + 2 y = - 156 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{21}{2} = -10\frac{1}{2} = -10.5$$

$$y = -\frac{207}{4} = -51\frac{3}{4} = -51.75$$

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.

$$7x-2y=30,5x+2y=-156$$

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

$$7x-2y=30$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$7x=2y+30$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(2y+30\right)$$

Whakareatia \frac{1}{7} ki te 30+2y.

$$x=\frac{2}{7}y+\frac{30}{7}$$

Whakakapia te \frac{30+2y}{7} mō te x ki tērā atu whārite, 5x+2y=-156.

$$5\left(\frac{2}{7}y+\frac{30}{7}\right)+2y=-156$$

Whakareatia 5 ki te \frac{30+2y}{7}.

$$\frac{10}{7}y+\frac{150}{7}+2y=-156$$

Tāpiri \frac{10y}{7} ki te 2y.

$$\frac{24}{7}y+\frac{150}{7}=-156$$

Me tango \frac{150}{7} mai i ngā taha e rua o te whārite.

$$\frac{24}{7}y=-\frac{1242}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{24}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{207}{4}$$

Whakaurua te -\frac{207}{4} mō y ki x=\frac{2}{7}y+\frac{30}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{7}\left(-\frac{207}{4}\right)+\frac{30}{7}$$

Whakareatia \frac{2}{7} ki te -\frac{207}{4} 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.

$$x=-\frac{207}{14}+\frac{30}{7}$$

Tāpiri \frac{30}{7} ki te -\frac{207}{14} 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.

$$x=-\frac{21}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{21}{2},y=-\frac{207}{4}$$

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.

$$7x-2y=30,5x+2y=-156$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-2\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\-156\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-2\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-2\\5&2\end{matrix}\right))\left(\begin{matrix}7&-2\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\5&2\end{matrix}\right))\left(\begin{matrix}30\\-156\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\5&2\end{matrix}\right))\left(\begin{matrix}30\\-156\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-2\\5&2\end{matrix}\right))\left(\begin{matrix}30\\-156\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-\left(-2\times 5\right)}&-\frac{-2}{7\times 2-\left(-2\times 5\right)}\\-\frac{5}{7\times 2-\left(-2\times 5\right)}&\frac{7}{7\times 2-\left(-2\times 5\right)}\end{matrix}\right)\left(\begin{matrix}30\\-156\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{1}{12}\\-\frac{5}{24}&\frac{7}{24}\end{matrix}\right)\left(\begin{matrix}30\\-156\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 30+\frac{1}{12}\left(-156\right)\\-\frac{5}{24}\times 30+\frac{7}{24}\left(-156\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{21}{2}\\-\frac{207}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{21}{2},y=-\frac{207}{4}$$

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.

$$7x-2y=30,5x+2y=-156$$

Kia ōrite ai a 7x me 5x, 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 7.

$$5\times 7x+5\left(-2\right)y=5\times 30,7\times 5x+7\times 2y=7\left(-156\right)$$

Whakarūnātia.

$$35x-10y=150,35x+14y=-1092$$

Me tango 35x+14y=-1092 mai i 35x-10y=150 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x-10y-14y=150+1092$$

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

$$-10y-14y=150+1092$$

Tāpiri -10y ki te -14y.

$$-24y=150+1092$$

Tāpiri 150 ki te 1092.

$$-24y=1242$$

Whakawehea ngā taha e rua ki te -24.

$$y=-\frac{207}{4}$$

Whakaurua te -\frac{207}{4} mō y ki 5x+2y=-156. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2\left(-\frac{207}{4}\right)=-156$$

Whakareatia 2 ki te -\frac{207}{4}.

$$5x-\frac{207}{2}=-156$$

Me tāpiri \frac{207}{2} ki ngā taha e rua o te whārite.

$$5x=-\frac{105}{2}$$

Whakawehea ngā taha e rua ki te 5.

$$x=-\frac{21}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{21}{2},y=-\frac{207}{4}$$