Whakaoti i te {l}{4x+5y=6}{6x-7y=-20}

$$\left. \begin{array} { l } { 4 x + 5 y = 6 } \\ { 6 x - 7 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=2$$

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.

$$4x+5y=6,6x-7y=-20$$

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

$$4x+5y=6$$

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

$$4x=-5y+6$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+6\right)$$

Whakareatia \frac{1}{4} ki te -5y+6.

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

Whakakapia te -\frac{5y}{4}+\frac{3}{2} mō te x ki tērā atu whārite, 6x-7y=-20.

$$6\left(-\frac{5}{4}y+\frac{3}{2}\right)-7y=-20$$

Whakareatia 6 ki te -\frac{5y}{4}+\frac{3}{2}.

$$-\frac{15}{2}y+9-7y=-20$$

Tāpiri -\frac{15y}{2} ki te -7y.

$$-\frac{29}{2}y+9=-20$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$-\frac{29}{2}y=-29$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

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

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

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

$$x=\frac{-5+3}{2}$$

Tāpiri \frac{3}{2} ki te -\frac{5}{2} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$

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.

$$4x+5y=6,6x-7y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\6&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\6&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\6&-7\end{matrix}\right))\left(\begin{matrix}4&5\\6&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&-7\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&-7\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\6&-7\end{matrix}\right))\left(\begin{matrix}6\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4\left(-7\right)-5\times 6}&-\frac{5}{4\left(-7\right)-5\times 6}\\-\frac{6}{4\left(-7\right)-5\times 6}&\frac{4}{4\left(-7\right)-5\times 6}\end{matrix}\right)\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{58}&\frac{5}{58}\\\frac{3}{29}&-\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}6\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{58}\times 6+\frac{5}{58}\left(-20\right)\\\frac{3}{29}\times 6-\frac{2}{29}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=2$$

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.

$$4x+5y=6,6x-7y=-20$$

Kia ōrite ai a 4x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$6\times 4x+6\times 5y=6\times 6,4\times 6x+4\left(-7\right)y=4\left(-20\right)$$

Whakarūnātia.

$$24x+30y=36,24x-28y=-80$$

Me tango 24x-28y=-80 mai i 24x+30y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x+30y+28y=36+80$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$30y+28y=36+80$$

Tāpiri 30y ki te 28y.

$$58y=36+80$$

Tāpiri 36 ki te 80.

$$58y=116$$

Whakawehea ngā taha e rua ki te 58.

$$y=2$$

Whakaurua te 2 mō y ki 6x-7y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-7\times 2=-20$$

Whakareatia -7 ki te 2.

$$6x-14=-20$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$6x=-6$$

Whakawehea ngā taha e rua ki te 6.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=2$$