Whakaoti i te {l}{3x+4y=23}{5x+4y=33}

$$\left. \begin{array} { l } { 3 x + 4 y = 23 } \\ { 5 x + 4 y = 33 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=2$$

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.

$$3x+4y=23,5x+4y=33$$

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

$$3x+4y=23$$

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

$$3x=-4y+23$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+23\right)$$

Whakareatia \frac{1}{3} ki te -4y+23.

$$x=-\frac{4}{3}y+\frac{23}{3}$$

Whakakapia te \frac{-4y+23}{3} mō te x ki tērā atu whārite, 5x+4y=33.

$$5\left(-\frac{4}{3}y+\frac{23}{3}\right)+4y=33$$

Whakareatia 5 ki te \frac{-4y+23}{3}.

$$-\frac{20}{3}y+\frac{115}{3}+4y=33$$

Tāpiri -\frac{20y}{3} ki te 4y.

$$-\frac{8}{3}y+\frac{115}{3}=33$$

Me tango \frac{115}{3} mai i ngā taha e rua o te whārite.

$$-\frac{8}{3}y=-\frac{16}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{3}, 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{4}{3}y+\frac{23}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\times 2+\frac{23}{3}$$

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

$$x=\frac{-8+23}{3}$$

Tāpiri \frac{23}{3} ki te -\frac{8}{3} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$

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.

$$3x+4y=23,5x+4y=33$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}23\\33\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\5&4\end{matrix}\right))\left(\begin{matrix}3&4\\5&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&4\end{matrix}\right))\left(\begin{matrix}23\\33\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&4\end{matrix}\right))\left(\begin{matrix}23\\33\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&4\end{matrix}\right))\left(\begin{matrix}23\\33\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{3\times 4-4\times 5}&-\frac{4}{3\times 4-4\times 5}\\-\frac{5}{3\times 4-4\times 5}&\frac{3}{3\times 4-4\times 5}\end{matrix}\right)\left(\begin{matrix}23\\33\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{5}{8}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}23\\33\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 23+\frac{1}{2}\times 33\\\frac{5}{8}\times 23-\frac{3}{8}\times 33\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=2$$

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.

$$3x+4y=23,5x+4y=33$$

Me tango 5x+4y=33 mai i 3x+4y=23 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-5x+4y-4y=23-33$$

Tāpiri 4y ki te -4y. Ka whakakore atu ngā kupu 4y me -4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3x-5x=23-33$$

Tāpiri 3x ki te -5x.

$$-2x=23-33$$

Tāpiri 23 ki te -33.

$$-2x=-10$$

Whakawehea ngā taha e rua ki te -2.

$$x=5$$

Whakaurua te 5 mō x ki 5x+4y=33. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$5\times 5+4y=33$$

Whakareatia 5 ki te 5.

$$25+4y=33$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$4y=8$$

Whakawehea ngā taha e rua ki te 4.

$$y=2$$

Kua oti te pūnaha te whakatau.

$$x=5,y=2$$