Whakaoti i te {l}{10x+2y=50}{7x+2y=20}

$$\left. \begin{array} { l } { 10 x + 2 y = 50 } \\ { 7 x + 2 y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=10$$

$$y=-25$$

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.

$$10x+2y=50,7x+2y=20$$

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

$$10x+2y=50$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$10x=-2y+50$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(-2y+50\right)$$

Whakareatia \frac{1}{10} ki te -2y+50.

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

Whakakapia te -\frac{y}{5}+5 mō te x ki tērā atu whārite, 7x+2y=20.

$$7\left(-\frac{1}{5}y+5\right)+2y=20$$

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

$$-\frac{7}{5}y+35+2y=20$$

Tāpiri -\frac{7y}{5} ki te 2y.

$$\frac{3}{5}y+35=20$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$\frac{3}{5}y=-15$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-25$$

Whakaurua te -25 mō y ki x=-\frac{1}{5}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\left(-25\right)+5$$

Whakareatia -\frac{1}{5} ki te -25.

$$x=5+5$$

Tāpiri 5 ki te 5.

$$x=10$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-25$$

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.

$$10x+2y=50,7x+2y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&2\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&2\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&2\\7&2\end{matrix}\right))\left(\begin{matrix}10&2\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\7&2\end{matrix}\right))\left(\begin{matrix}50\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\7&2\end{matrix}\right))\left(\begin{matrix}50\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&2\\7&2\end{matrix}\right))\left(\begin{matrix}50\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{10\times 2-2\times 7}&-\frac{2}{10\times 2-2\times 7}\\-\frac{7}{10\times 2-2\times 7}&\frac{10}{10\times 2-2\times 7}\end{matrix}\right)\left(\begin{matrix}50\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\-\frac{7}{6}&\frac{5}{3}\end{matrix}\right)\left(\begin{matrix}50\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 50-\frac{1}{3}\times 20\\-\frac{7}{6}\times 50+\frac{5}{3}\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\-25\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=10,y=-25$$

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.

$$10x+2y=50,7x+2y=20$$

Me tango 7x+2y=20 mai i 10x+2y=50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-7x+2y-2y=50-20$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10x-7x=50-20$$

Tāpiri 10x ki te -7x.

$$3x=50-20$$

Tāpiri 50 ki te -20.

$$3x=30$$

Whakawehea ngā taha e rua ki te 3.

$$x=10$$

Whakaurua te 10 mō x ki 7x+2y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$7\times 10+2y=20$$

Whakareatia 7 ki te 10.

$$70+2y=20$$

Me tango 70 mai i ngā taha e rua o te whārite.

$$2y=-50$$

Whakawehea ngā taha e rua ki te 2.

$$y=-25$$

Kua oti te pūnaha te whakatau.

$$x=10,y=-25$$