Whakaoti i te {l}{x-3/5=y-7/2}{11x=13y}

$$\left. \begin{array} { l } { \frac { x - 3 } { 5 } = \frac { y - 7 } { 2 } } \\ { 11 x = 13 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=13$$

$$y=11$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-3\right)=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-7.

$$2x-6=5y-35$$

Tangohia te 5y mai i ngā taha e rua.

$$2x-6-5y=-35$$

Me tāpiri te 6 ki ngā taha e rua.

$$2x-5y=-35+6$$

Tāpirihia te -35 ki te 6, ka -29.

$$2x-5y=-29$$

Whakaarohia te whārite tuarua. Tangohia te 13y mai i ngā taha e rua.

$$11x-13y=0$$

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.

$$2x-5y=-29,11x-13y=0$$

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

$$2x-5y=-29$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$2x=5y-29$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(5y-29\right)$$

Whakareatia \frac{1}{2} ki te 5y-29.

$$x=\frac{5}{2}y-\frac{29}{2}$$

Whakakapia te \frac{5y-29}{2} mō te x ki tērā atu whārite, 11x-13y=0.

$$11\left(\frac{5}{2}y-\frac{29}{2}\right)-13y=0$$

Whakareatia 11 ki te \frac{5y-29}{2}.

$$\frac{55}{2}y-\frac{319}{2}-13y=0$$

Tāpiri \frac{55y}{2} ki te -13y.

$$\frac{29}{2}y-\frac{319}{2}=0$$

Me tāpiri \frac{319}{2} ki ngā taha e rua o te whārite.

$$\frac{29}{2}y=\frac{319}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{29}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=11$$

Whakaurua te 11 mō y ki x=\frac{5}{2}y-\frac{29}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{2}\times 11-\frac{29}{2}$$

Whakareatia \frac{5}{2} ki te 11.

$$x=\frac{55-29}{2}$$

Tāpiri -\frac{29}{2} ki te \frac{55}{2} 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.

$$x=13$$

Kua oti te pūnaha te whakatau.

$$x=13,y=11$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-3\right)=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-7.

$$2x-6=5y-35$$

Tangohia te 5y mai i ngā taha e rua.

$$2x-6-5y=-35$$

Me tāpiri te 6 ki ngā taha e rua.

$$2x-5y=-35+6$$

Tāpirihia te -35 ki te 6, ka -29.

$$2x-5y=-29$$

Whakaarohia te whārite tuarua. Tangohia te 13y mai i ngā taha e rua.

$$11x-13y=0$$

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

$$2x-5y=-29,11x-13y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-29\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-5\\11&-13\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right))\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right))\left(\begin{matrix}-29\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right))\left(\begin{matrix}-29\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-5\\11&-13\end{matrix}\right))\left(\begin{matrix}-29\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{2\left(-13\right)-\left(-5\times 11\right)}&-\frac{-5}{2\left(-13\right)-\left(-5\times 11\right)}\\-\frac{11}{2\left(-13\right)-\left(-5\times 11\right)}&\frac{2}{2\left(-13\right)-\left(-5\times 11\right)}\end{matrix}\right)\left(\begin{matrix}-29\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{29}&\frac{5}{29}\\-\frac{11}{29}&\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}-29\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{29}\left(-29\right)\\-\frac{11}{29}\left(-29\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}13\\11\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=13,y=11$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2.

$$2\left(x-3\right)=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6=5\left(y-7\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-7.

$$2x-6=5y-35$$

Tangohia te 5y mai i ngā taha e rua.

$$2x-6-5y=-35$$

Me tāpiri te 6 ki ngā taha e rua.

$$2x-5y=-35+6$$

Tāpirihia te -35 ki te 6, ka -29.

$$2x-5y=-29$$

Whakaarohia te whārite tuarua. Tangohia te 13y mai i ngā taha e rua.

$$11x-13y=0$$

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.

$$2x-5y=-29,11x-13y=0$$

Kia ōrite ai a 2x me 11x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 11 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$11\times 2x+11\left(-5\right)y=11\left(-29\right),2\times 11x+2\left(-13\right)y=0$$

Whakarūnātia.

$$22x-55y=-319,22x-26y=0$$

Me tango 22x-26y=0 mai i 22x-55y=-319 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$22x-22x-55y+26y=-319$$

Tāpiri 22x ki te -22x. Ka whakakore atu ngā kupu 22x me -22x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-55y+26y=-319$$

Tāpiri -55y ki te 26y.

$$-29y=-319$$

Whakawehea ngā taha e rua ki te -29.

$$y=11$$

Whakaurua te 11 mō y ki 11x-13y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$11x-13\times 11=0$$

Whakareatia -13 ki te 11.

$$11x-143=0$$

Me tāpiri 143 ki ngā taha e rua o te whārite.

$$11x=143$$

Whakawehea ngā taha e rua ki te 11.

$$x=13$$

Kua oti te pūnaha te whakatau.

$$x=13,y=11$$