Whakaoti i te {l}{4x+y=15}{5y+19x=30}

$$\left. \begin{array} { l } { 4 x + y = 15 } \\ { 5 y + 19 x = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=45$$

$$y=-165$$

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.

$$4x+y=15,19x+5y=30$$

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

$$4x+y=15$$

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

$$4x=-y+15$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-y+15\right)$$

Whakareatia \frac{1}{4} ki te -y+15.

$$x=-\frac{1}{4}y+\frac{15}{4}$$

Whakakapia te \frac{-y+15}{4} mō te x ki tērā atu whārite, 19x+5y=30.

$$19\left(-\frac{1}{4}y+\frac{15}{4}\right)+5y=30$$

Whakareatia 19 ki te \frac{-y+15}{4}.

$$-\frac{19}{4}y+\frac{285}{4}+5y=30$$

Tāpiri -\frac{19y}{4} ki te 5y.

$$\frac{1}{4}y+\frac{285}{4}=30$$

Me tango \frac{285}{4} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}y=-\frac{165}{4}$$

Me whakarea ngā taha e rua ki te 4.

$$y=-165$$

Whakaurua te -165 mō y ki x=-\frac{1}{4}y+\frac{15}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\left(-165\right)+\frac{15}{4}$$

Whakareatia -\frac{1}{4} ki te -165.

$$x=\frac{165+15}{4}$$

Tāpiri \frac{15}{4} ki te \frac{165}{4} 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.

$$x=45$$

Kua oti te pūnaha te whakatau.

$$x=45,y=-165$$

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.

$$4x+y=15,19x+5y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\19&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\19&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\19&5\end{matrix}\right))\left(\begin{matrix}4&1\\19&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\19&5\end{matrix}\right))\left(\begin{matrix}15\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\19&5\end{matrix}\right))\left(\begin{matrix}15\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\19&5\end{matrix}\right))\left(\begin{matrix}15\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-19}&-\frac{1}{4\times 5-19}\\-\frac{19}{4\times 5-19}&\frac{4}{4\times 5-19}\end{matrix}\right)\left(\begin{matrix}15\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5&-1\\-19&4\end{matrix}\right)\left(\begin{matrix}15\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\times 15-30\\-19\times 15+4\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\-165\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=45,y=-165$$

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.

$$4x+y=15,19x+5y=30$$

Kia ōrite ai a 4x me 19x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 19 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$19\times 4x+19y=19\times 15,4\times 19x+4\times 5y=4\times 30$$

Whakarūnātia.

$$76x+19y=285,76x+20y=120$$

Me tango 76x+20y=120 mai i 76x+19y=285 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$76x-76x+19y-20y=285-120$$

Tāpiri 76x ki te -76x. Ka whakakore atu ngā kupu 76x me -76x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$19y-20y=285-120$$

Tāpiri 19y ki te -20y.

$$-y=285-120$$

Tāpiri 285 ki te -120.

$$-y=165$$

Whakawehea ngā taha e rua ki te -1.

$$y=-165$$

Whakaurua te -165 mō y ki 19x+5y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$19x+5\left(-165\right)=30$$

Whakareatia 5 ki te -165.

$$19x-825=30$$

Me tāpiri 825 ki ngā taha e rua o te whārite.

$$19x=855$$

Whakawehea ngā taha e rua ki te 19.

$$x=45$$

Kua oti te pūnaha te whakatau.

$$x=45,y=-165$$