Whakaoti i te {l}{x/4+y/2=7}{x/3-y/4=2}

$$\left\{ \begin{array} { l } { \frac { x } { 4 } + \frac { y } { 2 } = 7 } \\ { \frac { x } { 3 } - \frac { y } { 4 } = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 4,2.

$$x+2y=28$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4x-3y=24$$

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.

$$x+2y=28,4x-3y=24$$

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

$$x+2y=28$$

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

$$x=-2y+28$$

Whakakapia te -2y+28 mō te x ki tērā atu whārite, 4x-3y=24.

$$4\left(-2y+28\right)-3y=24$$

Whakareatia 4 ki te -2y+28.

$$-8y+112-3y=24$$

Tāpiri -8y ki te -3y.

$$-11y+112=24$$

Me tango 112 mai i ngā taha e rua o te whārite.

$$-11y=-88$$

Whakawehea ngā taha e rua ki te -11.

$$y=8$$

Whakaurua te 8 mō y ki x=-2y+28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\times 8+28$$

Whakareatia -2 ki te 8.

$$x=-16+28$$

Tāpiri 28 ki te -16.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=8$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 4,2.

$$x+2y=28$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4x-3y=24$$

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

$$x+2y=28,4x-3y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&2\\4&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&2\\4&-3\end{matrix}\right))\left(\begin{matrix}1&2\\4&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\4&-3\end{matrix}\right))\left(\begin{matrix}28\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{-3-2\times 4}&-\frac{2}{-3-2\times 4}\\-\frac{4}{-3-2\times 4}&\frac{1}{-3-2\times 4}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}&\frac{2}{11}\\\frac{4}{11}&-\frac{1}{11}\end{matrix}\right)\left(\begin{matrix}28\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{11}\times 28+\frac{2}{11}\times 24\\\frac{4}{11}\times 28-\frac{1}{11}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=8$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 4,2.

$$x+2y=28$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4.

$$4x-3y=24$$

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.

$$x+2y=28,4x-3y=24$$

Kia ōrite ai a x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$4x+4\times 2y=4\times 28,4x-3y=24$$

Whakarūnātia.

$$4x+8y=112,4x-3y=24$$

Me tango 4x-3y=24 mai i 4x+8y=112 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+8y+3y=112-24$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y+3y=112-24$$

Tāpiri 8y ki te 3y.

$$11y=112-24$$

Tāpiri 112 ki te -24.

$$11y=88$$

Whakawehea ngā taha e rua ki te 11.

$$y=8$$

Whakaurua te 8 mō y ki 4x-3y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-3\times 8=24$$

Whakareatia -3 ki te 8.

$$4x-24=24$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$4x=48$$

Whakawehea ngā taha e rua ki te 4.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=8$$