Whakaoti i te {r}{5y+x=44}{y-x=4}

$$\left. \begin{array} { r } { 5 y + x = 44 } \\ { y - x = 4 } \end{array} \right.$$

Whakaoti mō y, x

$$x=4$$

$$y=8$$

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.

$$5y+x=44,y-x=4$$

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

$$5y+x=44$$

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

$$5y=-x+44$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{1}{5}\left(-x+44\right)$$

Whakareatia \frac{1}{5} ki te -x+44.

$$y=-\frac{1}{5}x+\frac{44}{5}$$

Whakakapia te \frac{-x+44}{5} mō te y ki tērā atu whārite, y-x=4.

$$-\frac{1}{5}x+\frac{44}{5}-x=4$$

Tāpiri -\frac{x}{5} ki te -x.

$$-\frac{6}{5}x+\frac{44}{5}=4$$

Me tango \frac{44}{5} mai i ngā taha e rua o te whārite.

$$-\frac{6}{5}x=-\frac{24}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{6}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=4$$

Whakaurua te 4 mō x ki y=-\frac{1}{5}x+\frac{44}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{1}{5}\times 4+\frac{44}{5}$$

Whakareatia -\frac{1}{5} ki te 4.

$$y=\frac{-4+44}{5}$$

Tāpiri \frac{44}{5} ki te -\frac{4}{5} 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.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=4$$

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.

$$5y+x=44,y-x=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}44\\4\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&1\\1&-1\end{matrix}\right))\left(\begin{matrix}5&1\\1&-1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\1&-1\end{matrix}\right))\left(\begin{matrix}44\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\1&-1\end{matrix}\right))\left(\begin{matrix}44\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\1&-1\end{matrix}\right))\left(\begin{matrix}44\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5\left(-1\right)-1}&-\frac{1}{5\left(-1\right)-1}\\-\frac{1}{5\left(-1\right)-1}&\frac{5}{5\left(-1\right)-1}\end{matrix}\right)\left(\begin{matrix}44\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{1}{6}&-\frac{5}{6}\end{matrix}\right)\left(\begin{matrix}44\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 44+\frac{1}{6}\times 4\\\frac{1}{6}\times 44-\frac{5}{6}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}8\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=8,x=4$$

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.

$$5y+x=44,y-x=4$$

Kia ōrite ai a 5y me y, 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 5.

$$5y+x=44,5y+5\left(-1\right)x=5\times 4$$

Whakarūnātia.

$$5y+x=44,5y-5x=20$$

Me tango 5y-5x=20 mai i 5y+x=44 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5y-5y+x+5x=44-20$$

Tāpiri 5y ki te -5y. Ka whakakore atu ngā kupu 5y me -5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$x+5x=44-20$$

Tāpiri x ki te 5x.

$$6x=44-20$$

Tāpiri 44 ki te -20.

$$6x=24$$

Whakawehea ngā taha e rua ki te 6.

$$x=4$$

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

$$y-4=4$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=4$$