Whakaoti i te {l}{5(x-3)/4-3(2y+1)/10=4-7(x+y+1)/8}{6x-5(2y-7)=21}

$$\left\{ \begin{array} { l } { \frac { 5 ( x - 3 ) } { 4 } - \frac { 3 ( 2 y + 1 ) } { 10 } = \frac { 4 - 7 ( x + y + 1 ) } { 8 } } \\ { 6 x - 5 ( 2 y - 7 ) = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{329}{229} = 1\frac{100}{229} \approx 1.436681223$$

$$y = \frac{518}{229} = 2\frac{60}{229} \approx 2.262008734$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 40, arā, te tauraro pātahi he tino iti rawa te kitea o 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te 10 ki te 5, ka 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te -4 ki te 3, ka -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Tangohia te 12 i te -150, ka -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Tangohia te 7 i te 4, ka -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Me tāpiri te 35x ki ngā taha e rua.

$$50x-162-24y+35x=-15-35y$$

Pahekotia te 50x me 35x, ka 85x.

$$85x-162-24y=-15-35y$$

Me tāpiri te 35y ki ngā taha e rua.

$$85x-162-24y+35y=-15$$

Pahekotia te -24y me 35y, ka 11y.

$$85x-162+11y=-15$$

Me tāpiri te 162 ki ngā taha e rua.

$$85x+11y=-15+162$$

Tāpirihia te -15 ki te 162, ka 147.

$$85x+11y=147$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2y-7.

$$6x-10y+35=21$$

Tangohia te 35 mai i ngā taha e rua.

$$6x-10y=21-35$$

Tangohia te 35 i te 21, ka -14.

$$6x-10y=-14$$

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.

$$85x+11y=147,6x-10y=-14$$

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

$$85x+11y=147$$

Me tango 11y mai i ngā taha e rua o te whārite.

$$85x=-11y+147$$

Whakawehea ngā taha e rua ki te 85.

$$x=\frac{1}{85}\left(-11y+147\right)$$

Whakareatia \frac{1}{85} ki te -11y+147.

$$x=-\frac{11}{85}y+\frac{147}{85}$$

Whakakapia te \frac{-11y+147}{85} mō te x ki tērā atu whārite, 6x-10y=-14.

$$6\left(-\frac{11}{85}y+\frac{147}{85}\right)-10y=-14$$

Whakareatia 6 ki te \frac{-11y+147}{85}.

$$-\frac{66}{85}y+\frac{882}{85}-10y=-14$$

Tāpiri -\frac{66y}{85} ki te -10y.

$$-\frac{916}{85}y+\frac{882}{85}=-14$$

Me tango \frac{882}{85} mai i ngā taha e rua o te whārite.

$$-\frac{916}{85}y=-\frac{2072}{85}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{916}{85}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{518}{229}$$

Whakaurua te \frac{518}{229} mō y ki x=-\frac{11}{85}y+\frac{147}{85}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{11}{85}\times \frac{518}{229}+\frac{147}{85}$$

Whakareatia -\frac{11}{85} ki te \frac{518}{229} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{5698}{19465}+\frac{147}{85}$$

Tāpiri \frac{147}{85} ki te -\frac{5698}{19465} 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.

$$x=\frac{329}{229}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{329}{229},y=\frac{518}{229}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 40, arā, te tauraro pātahi he tino iti rawa te kitea o 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te 10 ki te 5, ka 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te -4 ki te 3, ka -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Tangohia te 12 i te -150, ka -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Tangohia te 7 i te 4, ka -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Me tāpiri te 35x ki ngā taha e rua.

$$50x-162-24y+35x=-15-35y$$

Pahekotia te 50x me 35x, ka 85x.

$$85x-162-24y=-15-35y$$

Me tāpiri te 35y ki ngā taha e rua.

$$85x-162-24y+35y=-15$$

Pahekotia te -24y me 35y, ka 11y.

$$85x-162+11y=-15$$

Me tāpiri te 162 ki ngā taha e rua.

$$85x+11y=-15+162$$

Tāpirihia te -15 ki te 162, ka 147.

$$85x+11y=147$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2y-7.

$$6x-10y+35=21$$

Tangohia te 35 mai i ngā taha e rua.

$$6x-10y=21-35$$

Tangohia te 35 i te 21, ka -14.

$$6x-10y=-14$$

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

$$85x+11y=147,6x-10y=-14$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}85&11\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}85&11\\6&-10\end{matrix}\right).

$$inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}85&11\\6&-10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}85&11\\6&-10\end{matrix}\right))\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{85\left(-10\right)-11\times 6}&-\frac{11}{85\left(-10\right)-11\times 6}\\-\frac{6}{85\left(-10\right)-11\times 6}&\frac{85}{85\left(-10\right)-11\times 6}\end{matrix}\right)\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{458}&\frac{11}{916}\\\frac{3}{458}&-\frac{85}{916}\end{matrix}\right)\left(\begin{matrix}147\\-14\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{458}\times 147+\frac{11}{916}\left(-14\right)\\\frac{3}{458}\times 147-\frac{85}{916}\left(-14\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{329}{229}\\\frac{518}{229}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{329}{229},y=\frac{518}{229}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 40, arā, te tauraro pātahi he tino iti rawa te kitea o 4,10,8.

$$10\times 5\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te 10 ki te 5, ka 50.

$$50\left(x-3\right)-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50 ki te x-3.

$$50x-150-4\times 3\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakareatia te -4 ki te 3, ka -12.

$$50x-150-12\left(2y+1\right)=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -12 ki te 2y+1.

$$50x-150-24y-12=5\left(4-7\left(x+y+1\right)\right)$$

Tangohia te 12 i te -150, ka -162.

$$50x-162-24y=5\left(4-7\left(x+y+1\right)\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te x+y+1.

$$50x-162-24y=5\left(4-7x-7y-7\right)$$

Tangohia te 7 i te 4, ka -3.

$$50x-162-24y=5\left(-3-7x-7y\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te -3-7x-7y.

$$50x-162-24y=-15-35x-35y$$

Me tāpiri te 35x ki ngā taha e rua.

$$50x-162-24y+35x=-15-35y$$

Pahekotia te 50x me 35x, ka 85x.

$$85x-162-24y=-15-35y$$

Me tāpiri te 35y ki ngā taha e rua.

$$85x-162-24y+35y=-15$$

Pahekotia te -24y me 35y, ka 11y.

$$85x-162+11y=-15$$

Me tāpiri te 162 ki ngā taha e rua.

$$85x+11y=-15+162$$

Tāpirihia te -15 ki te 162, ka 147.

$$85x+11y=147$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2y-7.

$$6x-10y+35=21$$

Tangohia te 35 mai i ngā taha e rua.

$$6x-10y=21-35$$

Tangohia te 35 i te 21, ka -14.

$$6x-10y=-14$$

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.

$$85x+11y=147,6x-10y=-14$$

Kia ōrite ai a 85x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 85.

$$6\times 85x+6\times 11y=6\times 147,85\times 6x+85\left(-10\right)y=85\left(-14\right)$$

Whakarūnātia.

$$510x+66y=882,510x-850y=-1190$$

Me tango 510x-850y=-1190 mai i 510x+66y=882 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$510x-510x+66y+850y=882+1190$$

Tāpiri 510x ki te -510x. Ka whakakore atu ngā kupu 510x me -510x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$66y+850y=882+1190$$

Tāpiri 66y ki te 850y.

$$916y=882+1190$$

Tāpiri 882 ki te 1190.

$$916y=2072$$

Whakawehea ngā taha e rua ki te 916.

$$y=\frac{518}{229}$$

Whakaurua te \frac{518}{229} mō y ki 6x-10y=-14. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-10\times \frac{518}{229}=-14$$

Whakareatia -10 ki te \frac{518}{229}.

$$6x-\frac{5180}{229}=-14$$

Me tāpiri \frac{5180}{229} ki ngā taha e rua o te whārite.

$$6x=\frac{1974}{229}$$

Whakawehea ngā taha e rua ki te 6.

$$x=\frac{329}{229}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{329}{229},y=\frac{518}{229}$$