Whakaoti i te {l}{4x+5y=3}{2x-3y=4}

$$\left. \begin{array} { l } { 4 x + 5 y = 3 } \\ { 2 x - 3 y = 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{29}{22} = 1\frac{7}{22} \approx 1.318181818$$

$$y=-\frac{5}{11}\approx -0.454545455$$

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.

$$4x+5y=3,2x-3y=4$$

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

$$4x+5y=3$$

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

$$4x=-5y+3$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te -5y+3.

$$x=-\frac{5}{4}y+\frac{3}{4}$$

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

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

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

$$-\frac{5}{2}y+\frac{3}{2}-3y=4$$

Tāpiri -\frac{5y}{2} ki te -3y.

$$-\frac{11}{2}y+\frac{3}{2}=4$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$-\frac{11}{2}y=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{11}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{5}{11}$$

Whakaurua te -\frac{5}{11} mō y ki x=-\frac{5}{4}y+\frac{3}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\left(-\frac{5}{11}\right)+\frac{3}{4}$$

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

$$x=\frac{25}{44}+\frac{3}{4}$$

Tāpiri \frac{3}{4} ki te \frac{25}{44} 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.

$$x=\frac{29}{22}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{29}{22},y=-\frac{5}{11}$$

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.

$$4x+5y=3,2x-3y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\2&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\2&-3\end{matrix}\right))\left(\begin{matrix}4&5\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\2&-3\end{matrix}\right))\left(\begin{matrix}3\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4\left(-3\right)-5\times 2}&-\frac{5}{4\left(-3\right)-5\times 2}\\-\frac{2}{4\left(-3\right)-5\times 2}&\frac{4}{4\left(-3\right)-5\times 2}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}&\frac{5}{22}\\\frac{1}{11}&-\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}3\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}\times 3+\frac{5}{22}\times 4\\\frac{1}{11}\times 3-\frac{2}{11}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{29}{22}\\-\frac{5}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{29}{22},y=-\frac{5}{11}$$

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.

$$4x+5y=3,2x-3y=4$$

Kia ōrite ai a 4x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$2\times 4x+2\times 5y=2\times 3,4\times 2x+4\left(-3\right)y=4\times 4$$

Whakarūnātia.

$$8x+10y=6,8x-12y=16$$

Me tango 8x-12y=16 mai i 8x+10y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+10y+12y=6-16$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y+12y=6-16$$

Tāpiri 10y ki te 12y.

$$22y=6-16$$

Tāpiri 6 ki te -16.

$$22y=-10$$

Whakawehea ngā taha e rua ki te 22.

$$y=-\frac{5}{11}$$

Whakaurua te -\frac{5}{11} mō y ki 2x-3y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\left(-\frac{5}{11}\right)=4$$

Whakareatia -3 ki te -\frac{5}{11}.

$$2x+\frac{15}{11}=4$$

Me tango \frac{15}{11} mai i ngā taha e rua o te whārite.

$$2x=\frac{29}{11}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{29}{22}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{29}{22},y=-\frac{5}{11}$$