Whakaoti i te {l}{4x-y=10}{3x+2y=8}

$$\left\{ \begin{array} { l } { 4 x - y = 10 } \\ { 3 x + 2 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{28}{11} = 2\frac{6}{11} \approx 2.545454545$$

$$y=\frac{2}{11}\approx 0.181818182$$

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.

$$4x-y=10,3x+2y=8$$

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

$$4x-y=10$$

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

$$4x=y+10$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

Whakakapia te \frac{y}{4}+\frac{5}{2} mō te x ki tērā atu whārite, 3x+2y=8.

$$3\left(\frac{1}{4}y+\frac{5}{2}\right)+2y=8$$

Whakareatia 3 ki te \frac{y}{4}+\frac{5}{2}.

$$\frac{3}{4}y+\frac{15}{2}+2y=8$$

Tāpiri \frac{3y}{4} ki te 2y.

$$\frac{11}{4}y+\frac{15}{2}=8$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$\frac{11}{4}y=\frac{1}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{2}{11}$$

Whakaurua te \frac{2}{11} mō y ki x=\frac{1}{4}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\times \frac{2}{11}+\frac{5}{2}$$

Whakareatia \frac{1}{4} ki te \frac{2}{11} 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.

$$x=\frac{1}{22}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{1}{22} 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.

$$x=\frac{28}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{28}{11},y=\frac{2}{11}$$

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.

$$4x-y=10,3x+2y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\3&2\end{matrix}\right))\left(\begin{matrix}4&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&2\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&2\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&2\end{matrix}\right))\left(\begin{matrix}10\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-3\right)}&-\frac{-1}{4\times 2-\left(-3\right)}\\-\frac{3}{4\times 2-\left(-3\right)}&\frac{4}{4\times 2-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}10\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}&\frac{1}{11}\\-\frac{3}{11}&\frac{4}{11}\end{matrix}\right)\left(\begin{matrix}10\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{11}\times 10+\frac{1}{11}\times 8\\-\frac{3}{11}\times 10+\frac{4}{11}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{28}{11}\\\frac{2}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{28}{11},y=\frac{2}{11}$$

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.

$$4x-y=10,3x+2y=8$$

Kia ōrite ai a 4x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$3\times 4x+3\left(-1\right)y=3\times 10,4\times 3x+4\times 2y=4\times 8$$

Whakarūnātia.

$$12x-3y=30,12x+8y=32$$

Me tango 12x+8y=32 mai i 12x-3y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-3y-8y=30-32$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-8y=30-32$$

Tāpiri -3y ki te -8y.

$$-11y=30-32$$

Tāpiri 30 ki te -32.

$$-11y=-2$$

Whakawehea ngā taha e rua ki te -11.

$$y=\frac{2}{11}$$

Whakaurua te \frac{2}{11} mō y ki 3x+2y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{2}{11}=8$$

Whakareatia 2 ki te \frac{2}{11}.

$$3x+\frac{4}{11}=8$$

Me tango \frac{4}{11} mai i ngā taha e rua o te whārite.

$$3x=\frac{84}{11}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{28}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{28}{11},y=\frac{2}{11}$$