Whakaoti i te {l}{3x+4y=253}{y=5x}

$$\left. \begin{array} { l } { 3 x + 4 y = 253 } \\ { y = 5 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=11$$

$$y=55$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

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.

$$3x+4y=253,-5x+y=0$$

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

$$3x+4y=253$$

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

$$3x=-4y+253$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+253\right)$$

Whakareatia \frac{1}{3} ki te -4y+253.

$$x=-\frac{4}{3}y+\frac{253}{3}$$

Whakakapia te \frac{-4y+253}{3} mō te x ki tērā atu whārite, -5x+y=0.

$$-5\left(-\frac{4}{3}y+\frac{253}{3}\right)+y=0$$

Whakareatia -5 ki te \frac{-4y+253}{3}.

$$\frac{20}{3}y-\frac{1265}{3}+y=0$$

Tāpiri \frac{20y}{3} ki te y.

$$\frac{23}{3}y-\frac{1265}{3}=0$$

Me tāpiri \frac{1265}{3} ki ngā taha e rua o te whārite.

$$\frac{23}{3}y=\frac{1265}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{23}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=55$$

Whakaurua te 55 mō y ki x=-\frac{4}{3}y+\frac{253}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\times 55+\frac{253}{3}$$

Whakareatia -\frac{4}{3} ki te 55.

$$x=\frac{-220+253}{3}$$

Tāpiri \frac{253}{3} ki te -\frac{220}{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.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=55$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x+4y=253,-5x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}253\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&4\\-5&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\-5&1\end{matrix}\right))\left(\begin{matrix}3&4\\-5&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\-5&1\end{matrix}\right))\left(\begin{matrix}253\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\-5&1\end{matrix}\right))\left(\begin{matrix}253\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\-5&1\end{matrix}\right))\left(\begin{matrix}253\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-4\left(-5\right)}&-\frac{4}{3-4\left(-5\right)}\\-\frac{-5}{3-4\left(-5\right)}&\frac{3}{3-4\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}253\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{23}&-\frac{4}{23}\\\frac{5}{23}&\frac{3}{23}\end{matrix}\right)\left(\begin{matrix}253\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{23}\times 253\\\frac{5}{23}\times 253\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}11\\55\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=11,y=55$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=0$$

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.

$$3x+4y=253,-5x+y=0$$

Kia ōrite ai a 3x me -5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-5\times 3x-5\times 4y=-5\times 253,3\left(-5\right)x+3y=0$$

Whakarūnātia.

$$-15x-20y=-1265,-15x+3y=0$$

Me tango -15x+3y=0 mai i -15x-20y=-1265 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-15x+15x-20y-3y=-1265$$

Tāpiri -15x ki te 15x. Ka whakakore atu ngā kupu -15x me 15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-20y-3y=-1265$$

Tāpiri -20y ki te -3y.

$$-23y=-1265$$

Whakawehea ngā taha e rua ki te -23.

$$y=55$$

Whakaurua te 55 mō y ki -5x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+55=0$$

Me tango 55 mai i ngā taha e rua o te whārite.

$$-5x=-55$$

Whakawehea ngā taha e rua ki te -5.

$$x=11$$

Kua oti te pūnaha te whakatau.

$$x=11,y=55$$