Whakaoti i te {l}{-3x+15y=59}{3x+4y=17}

$$\left. \begin{array} { l } { - 3 x + 15 y = 59 } \\ { 3 x + 4 y = 17 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{3}\approx 0.333333333$$

$$y=4$$

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.

$$-3x+15y=59,3x+4y=17$$

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

$$-3x+15y=59$$

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

$$-3x=-15y+59$$

Whakawehea ngā taha e rua ki te -3.

$$x=-\frac{1}{3}\left(-15y+59\right)$$

Whakareatia -\frac{1}{3} ki te -15y+59.

$$x=5y-\frac{59}{3}$$

Whakakapia te 5y-\frac{59}{3} mō te x ki tērā atu whārite, 3x+4y=17.

$$3\left(5y-\frac{59}{3}\right)+4y=17$$

Whakareatia 3 ki te 5y-\frac{59}{3}.

$$15y-59+4y=17$$

Tāpiri 15y ki te 4y.

$$19y-59=17$$

Me tāpiri 59 ki ngā taha e rua o te whārite.

$$19y=76$$

Whakawehea ngā taha e rua ki te 19.

$$y=4$$

Whakaurua te 4 mō y ki x=5y-\frac{59}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=5\times 4-\frac{59}{3}$$

Whakareatia 5 ki te 4.

$$x=20-\frac{59}{3}$$

Tāpiri -\frac{59}{3} ki te 20.

$$x=\frac{1}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{3},y=4$$

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.

$$-3x+15y=59,3x+4y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-3&15\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}59\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-3&15\\3&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-3&15\\3&4\end{matrix}\right))\left(\begin{matrix}-3&15\\3&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&15\\3&4\end{matrix}\right))\left(\begin{matrix}59\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&15\\3&4\end{matrix}\right))\left(\begin{matrix}59\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-3&15\\3&4\end{matrix}\right))\left(\begin{matrix}59\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{-3\times 4-15\times 3}&-\frac{15}{-3\times 4-15\times 3}\\-\frac{3}{-3\times 4-15\times 3}&-\frac{3}{-3\times 4-15\times 3}\end{matrix}\right)\left(\begin{matrix}59\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{57}&\frac{5}{19}\\\frac{1}{19}&\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}59\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{57}\times 59+\frac{5}{19}\times 17\\\frac{1}{19}\times 59+\frac{1}{19}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{3},y=4$$

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.

$$-3x+15y=59,3x+4y=17$$

Kia ōrite ai a -3x 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 -3.

$$3\left(-3\right)x+3\times 15y=3\times 59,-3\times 3x-3\times 4y=-3\times 17$$

Whakarūnātia.

$$-9x+45y=177,-9x-12y=-51$$

Me tango -9x-12y=-51 mai i -9x+45y=177 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+45y+12y=177+51$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$45y+12y=177+51$$

Tāpiri 45y ki te 12y.

$$57y=177+51$$

Tāpiri 177 ki te 51.

$$57y=228$$

Whakawehea ngā taha e rua ki te 57.

$$y=4$$

Whakaurua te 4 mō y ki 3x+4y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\times 4=17$$

Whakareatia 4 ki te 4.

$$3x+16=17$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$3x=1$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{3},y=4$$