Whakaoti i te {c}{8x+4y=10}{4x+9y=30}

$$\left. \begin{array} { c } { 8 x + 4 y = 10 } \\ { 4 x + 9 y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{15}{28}\approx -0.535714286$$

$$y = \frac{25}{7} = 3\frac{4}{7} \approx 3.571428571$$

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.

$$8x+4y=10,4x+9y=30$$

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

$$8x+4y=10$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$8x=-4y+10$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-4y+10\right)$$

Whakareatia \frac{1}{8} ki te -4y+10.

$$x=-\frac{1}{2}y+\frac{5}{4}$$

Whakakapia te -\frac{y}{2}+\frac{5}{4} mō te x ki tērā atu whārite, 4x+9y=30.

$$4\left(-\frac{1}{2}y+\frac{5}{4}\right)+9y=30$$

Whakareatia 4 ki te -\frac{y}{2}+\frac{5}{4}.

$$-2y+5+9y=30$$

Tāpiri -2y ki te 9y.

$$7y+5=30$$

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

$$7y=25$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{25}{7}$$

Whakaurua te \frac{25}{7} mō y ki x=-\frac{1}{2}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{25}{7}+\frac{5}{4}$$

Whakareatia -\frac{1}{2} ki te \frac{25}{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.

$$x=-\frac{25}{14}+\frac{5}{4}$$

Tāpiri \frac{5}{4} ki te -\frac{25}{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.

$$x=-\frac{15}{28}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{15}{28},y=\frac{25}{7}$$

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.

$$8x+4y=10,4x+9y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&4\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&4\\4&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&4\\4&9\end{matrix}\right))\left(\begin{matrix}8&4\\4&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\4&9\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\4&9\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&4\\4&9\end{matrix}\right))\left(\begin{matrix}10\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{8\times 9-4\times 4}&-\frac{4}{8\times 9-4\times 4}\\-\frac{4}{8\times 9-4\times 4}&\frac{8}{8\times 9-4\times 4}\end{matrix}\right)\left(\begin{matrix}10\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{56}&-\frac{1}{14}\\-\frac{1}{14}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}10\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{56}\times 10-\frac{1}{14}\times 30\\-\frac{1}{14}\times 10+\frac{1}{7}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{28}\\\frac{25}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{15}{28},y=\frac{25}{7}$$

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.

$$8x+4y=10,4x+9y=30$$

Kia ōrite ai a 8x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$4\times 8x+4\times 4y=4\times 10,8\times 4x+8\times 9y=8\times 30$$

Whakarūnātia.

$$32x+16y=40,32x+72y=240$$

Me tango 32x+72y=240 mai i 32x+16y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32x-32x+16y-72y=40-240$$

Tāpiri 32x ki te -32x. Ka whakakore atu ngā kupu 32x me -32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16y-72y=40-240$$

Tāpiri 16y ki te -72y.

$$-56y=40-240$$

Tāpiri 40 ki te -240.

$$-56y=-200$$

Whakawehea ngā taha e rua ki te -56.

$$y=\frac{25}{7}$$

Whakaurua te \frac{25}{7} mō y ki 4x+9y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+9\times \frac{25}{7}=30$$

Whakareatia 9 ki te \frac{25}{7}.

$$4x+\frac{225}{7}=30$$

Me tango \frac{225}{7} mai i ngā taha e rua o te whārite.

$$4x=-\frac{15}{7}$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{15}{28}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{15}{28},y=\frac{25}{7}$$