Whakaoti i te {l}{1/5x-2y=10}{3x-3/2y=36}

$$\left\{ \begin{array} { l } { \frac { 1 } { 5 } x - 2 y = 10 } \\ { 3 x - \frac { 3 } { 2 } y = 36 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-4$$

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.

$$\frac{1}{5}x-2y=10,3x-\frac{3}{2}y=36$$

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

$$\frac{1}{5}x-2y=10$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$\frac{1}{5}x=2y+10$$

Me whakarea ngā taha e rua ki te 5.

$$x=5\left(2y+10\right)$$

Whakareatia 5 ki te 10+2y.

$$x=10y+50$$

Whakakapia te 50+10y mō te x ki tērā atu whārite, 3x-\frac{3}{2}y=36.

$$3\left(10y+50\right)-\frac{3}{2}y=36$$

Whakareatia 3 ki te 50+10y.

$$30y+150-\frac{3}{2}y=36$$

Tāpiri 30y ki te -\frac{3y}{2}.

$$\frac{57}{2}y+150=36$$

Me tango 150 mai i ngā taha e rua o te whārite.

$$\frac{57}{2}y=-114$$

Whakawehea ngā taha e rua o te whārite ki te \frac{57}{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=10y+50. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=10\left(-4\right)+50$$

Whakareatia 10 ki te -4.

$$x=-40+50$$

Tāpiri 50 ki te -40.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-4$$

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.

$$\frac{1}{5}x-2y=10,3x-\frac{3}{2}y=36$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\36\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right))\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right))\left(\begin{matrix}10\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right))\left(\begin{matrix}10\\36\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{5}&-2\\3&-\frac{3}{2}\end{matrix}\right))\left(\begin{matrix}10\\36\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{3}{2}}{\frac{1}{5}\left(-\frac{3}{2}\right)-\left(-2\times 3\right)}&-\frac{-2}{\frac{1}{5}\left(-\frac{3}{2}\right)-\left(-2\times 3\right)}\\-\frac{3}{\frac{1}{5}\left(-\frac{3}{2}\right)-\left(-2\times 3\right)}&\frac{\frac{1}{5}}{\frac{1}{5}\left(-\frac{3}{2}\right)-\left(-2\times 3\right)}\end{matrix}\right)\left(\begin{matrix}10\\36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{19}&\frac{20}{57}\\-\frac{10}{19}&\frac{2}{57}\end{matrix}\right)\left(\begin{matrix}10\\36\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{19}\times 10+\frac{20}{57}\times 36\\-\frac{10}{19}\times 10+\frac{2}{57}\times 36\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=-4$$

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.

$$\frac{1}{5}x-2y=10,3x-\frac{3}{2}y=36$$

Kia ōrite ai a \frac{x}{5} me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te \frac{1}{5}.

$$3\times \frac{1}{5}x+3\left(-2\right)y=3\times 10,\frac{1}{5}\times 3x+\frac{1}{5}\left(-\frac{3}{2}\right)y=\frac{1}{5}\times 36$$

Whakarūnātia.

$$\frac{3}{5}x-6y=30,\frac{3}{5}x-\frac{3}{10}y=\frac{36}{5}$$

Me tango \frac{3}{5}x-\frac{3}{10}y=\frac{36}{5} mai i \frac{3}{5}x-6y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{5}x-\frac{3}{5}x-6y+\frac{3}{10}y=30-\frac{36}{5}$$

Tāpiri \frac{3x}{5} ki te -\frac{3x}{5}. Ka whakakore atu ngā kupu \frac{3x}{5} me -\frac{3x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+\frac{3}{10}y=30-\frac{36}{5}$$

Tāpiri -6y ki te \frac{3y}{10}.

$$-\frac{57}{10}y=30-\frac{36}{5}$$

Tāpiri 30 ki te -\frac{36}{5}.

$$-\frac{57}{10}y=\frac{114}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{57}{10}, 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 3x-\frac{3}{2}y=36. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-\frac{3}{2}\left(-4\right)=36$$

Whakareatia -\frac{3}{2} ki te -4.

$$3x+6=36$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x=30$$

Whakawehea ngā taha e rua ki te 3.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-4$$