Whakaoti i te {l}{5x+y=35}{7x+1,1y=40}

$$\left\{ \begin{array} { l } { 5 x + y = 35 } \\ { 7 x + 1,1 y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=30$$

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.

$$5x+y=35,7x+1.1y=40$$

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

$$5x+y=35$$

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

$$5x=-y+35$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-y+35\right)$$

Whakareatia \frac{1}{5} ki te -y+35.

$$x=-\frac{1}{5}y+7$$

Whakakapia te -\frac{y}{5}+7 mō te x ki tērā atu whārite, 7x+1.1y=40.

$$7\left(-\frac{1}{5}y+7\right)+1.1y=40$$

Whakareatia 7 ki te -\frac{y}{5}+7.

$$-\frac{7}{5}y+49+1.1y=40$$

Tāpiri -\frac{7y}{5} ki te \frac{11y}{10}.

$$-\frac{3}{10}y+49=40$$

Me tango 49 mai i ngā taha e rua o te whārite.

$$-\frac{3}{10}y=-9$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=30$$

Whakaurua te 30 mō y ki x=-\frac{1}{5}y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\times 30+7$$

Whakareatia -\frac{1}{5} ki te 30.

$$x=-6+7$$

Tāpiri 7 ki te -6.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=30$$

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.

$$5x+y=35,7x+1.1y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\40\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&1\\7&1.1\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right))\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&1\\7&1.1\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1.1}{5\times 1.1-7}&-\frac{1}{5\times 1.1-7}\\-\frac{7}{5\times 1.1-7}&\frac{5}{5\times 1.1-7}\end{matrix}\right)\left(\begin{matrix}35\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{15}&\frac{2}{3}\\\frac{14}{3}&-\frac{10}{3}\end{matrix}\right)\left(\begin{matrix}35\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{15}\times 35+\frac{2}{3}\times 40\\\frac{14}{3}\times 35-\frac{10}{3}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=30$$

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.

$$5x+y=35,7x+1.1y=40$$

Kia ōrite ai a 5x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$7\times 5x+7y=7\times 35,5\times 7x+5\times 1.1y=5\times 40$$

Whakarūnātia.

$$35x+7y=245,35x+5.5y=200$$

Me tango 35x+5.5y=200 mai i 35x+7y=245 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x+7y-5.5y=245-200$$

Tāpiri 35x ki te -35x. Ka whakakore atu ngā kupu 35x me -35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y-5.5y=245-200$$

Tāpiri 7y ki te -\frac{11y}{2}.

$$1.5y=245-200$$

Tāpiri 245 ki te -200.

$$1.5y=45$$

Whakawehea ngā taha e rua o te whārite ki te 1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=30$$

Whakaurua te 30 mō y ki 7x+1.1y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+1.1\times 30=40$$

Whakareatia 1.1 ki te 30.

$$7x+33=40$$

Me tango 33 mai i ngā taha e rua o te whārite.

$$7x=7$$

Whakawehea ngā taha e rua ki te 7.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=30$$