Whakaoti i te {l}{4x-5y=7}{2x+3y=1}

$$\left\{ \begin{array} { l } { 4 x - 5 y = 7 } \\ { 2 x + 3 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{13}{11} = 1\frac{2}{11} \approx 1.181818182$$

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

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.

$$4x-5y=7,2x+3y=1$$

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

$$4x-5y=7$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$4x=5y+7$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(5y+7\right)$$

Whakareatia \frac{1}{4} ki te 5y+7.

$$x=\frac{5}{4}y+\frac{7}{4}$$

Whakakapia te \frac{5y+7}{4} mō te x ki tērā atu whārite, 2x+3y=1.

$$2\left(\frac{5}{4}y+\frac{7}{4}\right)+3y=1$$

Whakareatia 2 ki te \frac{5y+7}{4}.

$$\frac{5}{2}y+\frac{7}{2}+3y=1$$

Tāpiri \frac{5y}{2} ki te 3y.

$$\frac{11}{2}y+\frac{7}{2}=1$$

Me tango \frac{7}{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{7}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4}\left(-\frac{5}{11}\right)+\frac{7}{4}$$

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.

$$x=-\frac{25}{44}+\frac{7}{4}$$

Tāpiri \frac{7}{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.

$$x=\frac{13}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{11},y=-\frac{5}{11}$$

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.

$$4x-5y=7,2x+3y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-5\\2&3\end{matrix}\right))\left(\begin{matrix}4&-5\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\2&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\2&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\2&3\end{matrix}\right))\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-\left(-5\times 2\right)}&-\frac{-5}{4\times 3-\left(-5\times 2\right)}\\-\frac{2}{4\times 3-\left(-5\times 2\right)}&\frac{4}{4\times 3-\left(-5\times 2\right)}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}&\frac{5}{22}\\-\frac{1}{11}&\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}7\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}\times 7+\frac{5}{22}\\-\frac{1}{11}\times 7+\frac{2}{11}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{11}\\-\frac{5}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{13}{11},y=-\frac{5}{11}$$

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.

$$4x-5y=7,2x+3y=1$$

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.

$$2\times 4x+2\left(-5\right)y=2\times 7,4\times 2x+4\times 3y=4$$

Whakarūnātia.

$$8x-10y=14,8x+12y=4$$

Me tango 8x+12y=4 mai i 8x-10y=14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-10y-12y=14-4$$

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.

$$-10y-12y=14-4$$

Tāpiri -10y ki te -12y.

$$-22y=14-4$$

Tāpiri 14 ki te -4.

$$-22y=10$$

Whakawehea ngā taha e rua ki te -22.

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

Whakaurua te -\frac{5}{11} mō y ki 2x+3y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+3\left(-\frac{5}{11}\right)=1$$

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

$$2x-\frac{15}{11}=1$$

Me tāpiri \frac{15}{11} ki ngā taha e rua o te whārite.

$$2x=\frac{26}{11}$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{13}{11}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{13}{11},y=-\frac{5}{11}$$