Whakaoti i te {l}{4x+y=100}{2x+2y=56}

$$\left. \begin{array} { l } { 4 x + y = 100 } \\ { 2 x + 2 y = 56 } \end{array} \right.$$

Whakaoti mō x, y

$$x=24$$

$$y=4$$

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.

$$4x+y=100,2x+2y=56$$

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

$$4x+y=100$$

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

$$4x=-y+100$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-y+100\right)$$

Whakareatia \frac{1}{4} ki te -y+100.

$$x=-\frac{1}{4}y+25$$

Whakakapia te -\frac{y}{4}+25 mō te x ki tērā atu whārite, 2x+2y=56.

$$2\left(-\frac{1}{4}y+25\right)+2y=56$$

Whakareatia 2 ki te -\frac{y}{4}+25.

$$-\frac{1}{2}y+50+2y=56$$

Tāpiri -\frac{y}{2} ki te 2y.

$$\frac{3}{2}y+50=56$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$\frac{3}{2}y=6$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=4$$

Whakaurua te 4 mō y ki x=-\frac{1}{4}y+25. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times 4+25$$

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

$$x=-1+25$$

Tāpiri 25 ki te -1.

$$x=24$$

Kua oti te pūnaha te whakatau.

$$x=24,y=4$$

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.

$$4x+y=100,2x+2y=56$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\56\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\2&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\2&2\end{matrix}\right))\left(\begin{matrix}4&1\\2&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&2\end{matrix}\right))\left(\begin{matrix}100\\56\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&2\end{matrix}\right))\left(\begin{matrix}100\\56\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&2\end{matrix}\right))\left(\begin{matrix}100\\56\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-2}&-\frac{1}{4\times 2-2}\\-\frac{2}{4\times 2-2}&\frac{4}{4\times 2-2}\end{matrix}\right)\left(\begin{matrix}100\\56\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{6}\\-\frac{1}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}100\\56\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 100-\frac{1}{6}\times 56\\-\frac{1}{3}\times 100+\frac{2}{3}\times 56\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=24,y=4$$

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.

$$4x+y=100,2x+2y=56$$

Kia ōrite ai a 4x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$2\times 4x+2y=2\times 100,4\times 2x+4\times 2y=4\times 56$$

Whakarūnātia.

$$8x+2y=200,8x+8y=224$$

Me tango 8x+8y=224 mai i 8x+2y=200 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+2y-8y=200-224$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y-8y=200-224$$

Tāpiri 2y ki te -8y.

$$-6y=200-224$$

Tāpiri 200 ki te -224.

$$-6y=-24$$

Whakawehea ngā taha e rua ki te -6.

$$y=4$$

Whakaurua te 4 mō y ki 2x+2y=56. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x+2\times 4=56$$

Whakareatia 2 ki te 4.

$$2x+8=56$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$2x=48$$

Whakawehea ngā taha e rua ki te 2.

$$x=24$$

Kua oti te pūnaha te whakatau.

$$x=24,y=4$$