Whakaoti i te {l}{10x+y=6y+5}{10y+x=(10x+y)+27}

$$\left\{ \begin{array} { l } { 10 x + y = 6 y + 5 } \\ { 10 y + x = ( 10 x + y ) + 27 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=7$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$10x+y-6y=5$$

Pahekotia te y me -6y, ka -5y.

$$10x-5y=5$$

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$10y+x-10x=y+27$$

Pahekotia te x me -10x, ka -9x.

$$10y-9x=y+27$$

Tangohia te y mai i ngā taha e rua.

$$10y-9x-y=27$$

Pahekotia te 10y me -y, ka 9y.

$$9y-9x=27$$

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.

$$10x-5y=5,-9x+9y=27$$

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

$$10x-5y=5$$

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

$$10x=5y+5$$

Whakawehea ngā taha e rua ki te 10.

$$x=\frac{1}{10}\left(5y+5\right)$$

Whakareatia \frac{1}{10} ki te 5+5y.

$$x=\frac{1}{2}y+\frac{1}{2}$$

Whakakapia te \frac{1+y}{2} mō te x ki tērā atu whārite, -9x+9y=27.

$$-9\left(\frac{1}{2}y+\frac{1}{2}\right)+9y=27$$

Whakareatia -9 ki te \frac{1+y}{2}.

$$-\frac{9}{2}y-\frac{9}{2}+9y=27$$

Tāpiri -\frac{9y}{2} ki te 9y.

$$\frac{9}{2}y-\frac{9}{2}=27$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$\frac{9}{2}y=\frac{63}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{1}{2}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times 7+\frac{1}{2}$$

Whakareatia \frac{1}{2} ki te 7.

$$x=\frac{7+1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{7}{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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$10x+y-6y=5$$

Pahekotia te y me -6y, ka -5y.

$$10x-5y=5$$

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$10y+x-10x=y+27$$

Pahekotia te x me -10x, ka -9x.

$$10y-9x=y+27$$

Tangohia te y mai i ngā taha e rua.

$$10y-9x-y=27$$

Pahekotia te 10y me -y, ka 9y.

$$9y-9x=27$$

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

$$10x-5y=5,-9x+9y=27$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\27\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}10&-5\\-9&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right))\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right))\left(\begin{matrix}5\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right))\left(\begin{matrix}5\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&-5\\-9&9\end{matrix}\right))\left(\begin{matrix}5\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{10\times 9-\left(-5\left(-9\right)\right)}&-\frac{-5}{10\times 9-\left(-5\left(-9\right)\right)}\\-\frac{-9}{10\times 9-\left(-5\left(-9\right)\right)}&\frac{10}{10\times 9-\left(-5\left(-9\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{9}\\\frac{1}{5}&\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}5\\27\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 5+\frac{1}{9}\times 27\\\frac{1}{5}\times 5+\frac{2}{9}\times 27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=7$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 6y mai i ngā taha e rua.

$$10x+y-6y=5$$

Pahekotia te y me -6y, ka -5y.

$$10x-5y=5$$

Whakaarohia te whārite tuarua. Tangohia te 10x mai i ngā taha e rua.

$$10y+x-10x=y+27$$

Pahekotia te x me -10x, ka -9x.

$$10y-9x=y+27$$

Tangohia te y mai i ngā taha e rua.

$$10y-9x-y=27$$

Pahekotia te 10y me -y, ka 9y.

$$9y-9x=27$$

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.

$$10x-5y=5,-9x+9y=27$$

Kia ōrite ai a 10x me -9x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -9 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 10.

$$-9\times 10x-9\left(-5\right)y=-9\times 5,10\left(-9\right)x+10\times 9y=10\times 27$$

Whakarūnātia.

$$-90x+45y=-45,-90x+90y=270$$

Me tango -90x+90y=270 mai i -90x+45y=-45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-90x+90x+45y-90y=-45-270$$

Tāpiri -90x ki te 90x. Ka whakakore atu ngā kupu -90x me 90x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$45y-90y=-45-270$$

Tāpiri 45y ki te -90y.

$$-45y=-45-270$$

Tāpiri -45 ki te -270.

$$-45y=-315$$

Whakawehea ngā taha e rua ki te -45.

$$y=7$$

Whakaurua te 7 mō y ki -9x+9y=27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-9x+9\times 7=27$$

Whakareatia 9 ki te 7.

$$-9x+63=27$$

Me tango 63 mai i ngā taha e rua o te whārite.

$$-9x=-36$$

Whakawehea ngā taha e rua ki te -9.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=7$$