Whakaoti i te {l}{x/6-y/5=-4}{x/4-y/6=-2}

$$\left\{ \begin{array} { l } { \frac { x } { 6 } - \frac { y } { 5 } = - 4 } \\ { \frac { x } { 4 } - \frac { y } { 6 } = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=30$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6y=-120$$

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 4,6.

$$3x-2y=-24$$

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.

$$5x-6y=-120,3x-2y=-24$$

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

$$5x-6y=-120$$

Me tāpiri 6y ki ngā taha e rua o te whārite.

$$5x=6y-120$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(6y-120\right)$$

Whakareatia \frac{1}{5} ki te -120+6y.

$$x=\frac{6}{5}y-24$$

Whakakapia te \frac{6y}{5}-24 mō te x ki tērā atu whārite, 3x-2y=-24.

$$3\left(\frac{6}{5}y-24\right)-2y=-24$$

Whakareatia 3 ki te \frac{6y}{5}-24.

$$\frac{18}{5}y-72-2y=-24$$

Tāpiri \frac{18y}{5} ki te -2y.

$$\frac{8}{5}y-72=-24$$

Me tāpiri 72 ki ngā taha e rua o te whārite.

$$\frac{8}{5}y=48$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{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 x=\frac{6}{5}y-24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{6}{5}\times 30-24$$

Whakareatia \frac{6}{5} ki te 30.

$$x=36-24$$

Tāpiri -24 ki te 36.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=30$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6y=-120$$

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 4,6.

$$3x-2y=-24$$

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

$$5x-6y=-120,3x-2y=-24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-6\\3&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right))\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right))\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right))\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-6\\3&-2\end{matrix}\right))\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{5\left(-2\right)-\left(-6\times 3\right)}&-\frac{-6}{5\left(-2\right)-\left(-6\times 3\right)}\\-\frac{3}{5\left(-2\right)-\left(-6\times 3\right)}&\frac{5}{5\left(-2\right)-\left(-6\times 3\right)}\end{matrix}\right)\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{3}{4}\\-\frac{3}{8}&\frac{5}{8}\end{matrix}\right)\left(\begin{matrix}-120\\-24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-120\right)+\frac{3}{4}\left(-24\right)\\-\frac{3}{8}\left(-120\right)+\frac{5}{8}\left(-24\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=30$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6y=-120$$

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 4,6.

$$3x-2y=-24$$

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.

$$5x-6y=-120,3x-2y=-24$$

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3\left(-6\right)y=3\left(-120\right),5\times 3x+5\left(-2\right)y=5\left(-24\right)$$

Whakarūnātia.

$$15x-18y=-360,15x-10y=-120$$

Me tango 15x-10y=-120 mai i 15x-18y=-360 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-18y+10y=-360+120$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-18y+10y=-360+120$$

Tāpiri -18y ki te 10y.

$$-8y=-360+120$$

Tāpiri -360 ki te 120.

$$-8y=-240$$

Whakawehea ngā taha e rua ki te -8.

$$y=30$$

Whakaurua te 30 mō y ki 3x-2y=-24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times 30=-24$$

Whakareatia -2 ki te 30.

$$3x-60=-24$$

Me tāpiri 60 ki ngā taha e rua o te whārite.

$$3x=36$$

Whakawehea ngā taha e rua ki te 3.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=30$$