Whakaoti i te {l}{3x+4y=190}{x-y=40}

$$\left\{ \begin{array} { l } { 3 x + 4 y = 190 } \\ { x - y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=50$$

$$y=10$$

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.

$$3x+4y=190,x-y=40$$

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

$$3x+4y=190$$

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

$$3x=-4y+190$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-4y+190\right)$$

Whakareatia \frac{1}{3} ki te -4y+190.

$$x=-\frac{4}{3}y+\frac{190}{3}$$

Whakakapia te \frac{-4y+190}{3} mō te x ki tērā atu whārite, x-y=40.

$$-\frac{4}{3}y+\frac{190}{3}-y=40$$

Tāpiri -\frac{4y}{3} ki te -y.

$$-\frac{7}{3}y+\frac{190}{3}=40$$

Me tango \frac{190}{3} mai i ngā taha e rua o te whārite.

$$-\frac{7}{3}y=-\frac{70}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki x=-\frac{4}{3}y+\frac{190}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{3}\times 10+\frac{190}{3}$$

Whakareatia -\frac{4}{3} ki te 10.

$$x=\frac{-40+190}{3}$$

Tāpiri \frac{190}{3} ki te -\frac{40}{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.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=10$$

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.

$$3x+4y=190,x-y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}190\\40\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\1&-1\end{matrix}\right))\left(\begin{matrix}3&4\\1&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-1\end{matrix}\right))\left(\begin{matrix}190\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-1\end{matrix}\right))\left(\begin{matrix}190\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\1&-1\end{matrix}\right))\left(\begin{matrix}190\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-4}&-\frac{4}{3\left(-1\right)-4}\\-\frac{1}{3\left(-1\right)-4}&\frac{3}{3\left(-1\right)-4}\end{matrix}\right)\left(\begin{matrix}190\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{4}{7}\\\frac{1}{7}&-\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}190\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 190+\frac{4}{7}\times 40\\\frac{1}{7}\times 190-\frac{3}{7}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=50,y=10$$

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.

$$3x+4y=190,x-y=40$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+4y=190,3x+3\left(-1\right)y=3\times 40$$

Whakarūnātia.

$$3x+4y=190,3x-3y=120$$

Me tango 3x-3y=120 mai i 3x+4y=190 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+4y+3y=190-120$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y+3y=190-120$$

Tāpiri 4y ki te 3y.

$$7y=190-120$$

Tāpiri 190 ki te -120.

$$7y=70$$

Whakawehea ngā taha e rua ki te 7.

$$y=10$$

Whakaurua te 10 mō y ki x-y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-10=40$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$x=50$$

Kua oti te pūnaha te whakatau.

$$x=50,y=10$$