Whakaoti i te {l}{5x+2y=25}{3x+4y=15}

$$\left\{ \begin{array} { l } { 5 x + 2 y = 25 } \\ { 3 x + 4 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=0$$

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.

$$5x+2y=25,3x+4y=15$$

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

$$5x+2y=25$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$5x=-2y+25$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+25\right)$$

Whakareatia \frac{1}{5} ki te -2y+25.

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

Whakakapia te -\frac{2y}{5}+5 mō te x ki tērā atu whārite, 3x+4y=15.

$$3\left(-\frac{2}{5}y+5\right)+4y=15$$

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

$$-\frac{6}{5}y+15+4y=15$$

Tāpiri -\frac{6y}{5} ki te 4y.

$$\frac{14}{5}y+15=15$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$\frac{14}{5}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{14}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{2}{5}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$

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.

$$5x+2y=25,3x+4y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&2\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\15\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&2\\3&4\end{matrix}\right))\left(\begin{matrix}5&2\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&4\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&4\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\3&4\end{matrix}\right))\left(\begin{matrix}25\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-2\times 3}&-\frac{2}{5\times 4-2\times 3}\\-\frac{3}{5\times 4-2\times 3}&\frac{5}{5\times 4-2\times 3}\end{matrix}\right)\left(\begin{matrix}25\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&-\frac{1}{7}\\-\frac{3}{14}&\frac{5}{14}\end{matrix}\right)\left(\begin{matrix}25\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 25-\frac{1}{7}\times 15\\-\frac{3}{14}\times 25+\frac{5}{14}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=0$$

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.

$$5x+2y=25,3x+4y=15$$

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

$$3\times 5x+3\times 2y=3\times 25,5\times 3x+5\times 4y=5\times 15$$

Whakarūnātia.

$$15x+6y=75,15x+20y=75$$

Me tango 15x+20y=75 mai i 15x+6y=75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+6y-20y=75-75$$

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.

$$6y-20y=75-75$$

Tāpiri 6y ki te -20y.

$$-14y=75-75$$

Tāpiri 75 ki te -75.

$$-14y=0$$

Whakawehea ngā taha e rua ki te -14.

$$y=0$$

Whakaurua te 0 mō y ki 3x+4y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=0$$