Whakaoti i te {l}{5x=2y+16}{7x=32-2y}

$$\left. \begin{array} { l } { 5 x = 2 y + 16 } \\ { 7 x = 32 - 2 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$5x-2y=16$$

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

$$7x+2y=32$$

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.

$$5x-2y=16,7x+2y=32$$

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

$$5x-2y=16$$

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

$$5x=2y+16$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(2y+16\right)$$

Whakareatia \frac{1}{5} ki te 16+2y.

$$x=\frac{2}{5}y+\frac{16}{5}$$

Whakakapia te \frac{16+2y}{5} mō te x ki tērā atu whārite, 7x+2y=32.

$$7\left(\frac{2}{5}y+\frac{16}{5}\right)+2y=32$$

Whakareatia 7 ki te \frac{16+2y}{5}.

$$\frac{14}{5}y+\frac{112}{5}+2y=32$$

Tāpiri \frac{14y}{5} ki te 2y.

$$\frac{24}{5}y+\frac{112}{5}=32$$

Me tango \frac{112}{5} mai i ngā taha e rua o te whārite.

$$\frac{24}{5}y=\frac{48}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{24}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{2}{5}y+\frac{16}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{5}\times 2+\frac{16}{5}$$

Whakareatia \frac{2}{5} ki te 2.

$$x=\frac{4+16}{5}$$

Tāpiri \frac{16}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$5x-2y=16$$

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

$$7x+2y=32$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$5x-2y=16,7x+2y=32$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-2\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\32\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-2\\7&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-2\\7&2\end{matrix}\right))\left(\begin{matrix}5&-2\\7&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\7&2\end{matrix}\right))\left(\begin{matrix}16\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\7&2\end{matrix}\right))\left(\begin{matrix}16\\32\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-2\\7&2\end{matrix}\right))\left(\begin{matrix}16\\32\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-2\times 7\right)}&-\frac{-2}{5\times 2-\left(-2\times 7\right)}\\-\frac{7}{5\times 2-\left(-2\times 7\right)}&\frac{5}{5\times 2-\left(-2\times 7\right)}\end{matrix}\right)\left(\begin{matrix}16\\32\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}&\frac{1}{12}\\-\frac{7}{24}&\frac{5}{24}\end{matrix}\right)\left(\begin{matrix}16\\32\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{12}\times 16+\frac{1}{12}\times 32\\-\frac{7}{24}\times 16+\frac{5}{24}\times 32\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2y mai i ngā taha e rua.

$$5x-2y=16$$

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

$$7x+2y=32$$

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.

$$5x-2y=16,7x+2y=32$$

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.

$$7\times 5x+7\left(-2\right)y=7\times 16,5\times 7x+5\times 2y=5\times 32$$

Whakarūnātia.

$$35x-14y=112,35x+10y=160$$

Me tango 35x+10y=160 mai i 35x-14y=112 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x-14y-10y=112-160$$

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.

$$-14y-10y=112-160$$

Tāpiri -14y ki te -10y.

$$-24y=112-160$$

Tāpiri 112 ki te -160.

$$-24y=-48$$

Whakawehea ngā taha e rua ki te -24.

$$y=2$$

Whakaurua te 2 mō y ki 7x+2y=32. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+2\times 2=32$$

Whakareatia 2 ki te 2.

$$7x+4=32$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$7x=28$$

Whakawehea ngā taha e rua ki te 7.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$