Whakaoti i te {l}{6-5y+2z/3=16+2y-3z/2}{6-5y+2z/3=1-5y+4z/4}

$$\left. \begin{array} { l } { \frac { 6 - 5 y + 2 z } { 3 } = \frac { 16 + 2 y - 3 z } { 2 } } \\ { \frac { 6 - 5 y + 2 z } { 3 } = \frac { 1 - 5 y + 4 z } { 4 } } \end{array} \right.$$

Whakaoti mō y, z

$$y=1$$

$$z=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(6-5y+2z\right)=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6-5y+2z.

$$12-10y+4z=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 16+2y-3z.

$$12-10y+4z=48+6y-9z$$

Tangohia te 6y mai i ngā taha e rua.

$$12-10y+4z-6y=48-9z$$

Pahekotia te -10y me -6y, ka -16y.

$$12-16y+4z=48-9z$$

Me tāpiri te 9z ki ngā taha e rua.

$$12-16y+4z+9z=48$$

Pahekotia te 4z me 9z, ka 13z.

$$12-16y+13z=48$$

Tangohia te 12 mai i ngā taha e rua.

$$-16y+13z=48-12$$

Tangohia te 12 i te 48, ka 36.

$$-16y+13z=36$$

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.

$$4\left(6-5y+2z\right)=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 6-5y+2z.

$$24-20y+8z=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1-5y+4z.

$$24-20y+8z=3-15y+12z$$

Me tāpiri te 15y ki ngā taha e rua.

$$24-20y+8z+15y=3+12z$$

Pahekotia te -20y me 15y, ka -5y.

$$24-5y+8z=3+12z$$

Tangohia te 12z mai i ngā taha e rua.

$$24-5y+8z-12z=3$$

Pahekotia te 8z me -12z, ka -4z.

$$24-5y-4z=3$$

Tangohia te 24 mai i ngā taha e rua.

$$-5y-4z=3-24$$

Tangohia te 24 i te 3, ka -21.

$$-5y-4z=-21$$

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.

$$-16y+13z=36,-5y-4z=-21$$

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

$$-16y+13z=36$$

Me tango 13z mai i ngā taha e rua o te whārite.

$$-16y=-13z+36$$

Whakawehea ngā taha e rua ki te -16.

$$y=-\frac{1}{16}\left(-13z+36\right)$$

Whakareatia -\frac{1}{16} ki te -13z+36.

$$y=\frac{13}{16}z-\frac{9}{4}$$

Whakakapia te \frac{13z}{16}-\frac{9}{4} mō te y ki tērā atu whārite, -5y-4z=-21.

$$-5\left(\frac{13}{16}z-\frac{9}{4}\right)-4z=-21$$

Whakareatia -5 ki te \frac{13z}{16}-\frac{9}{4}.

$$-\frac{65}{16}z+\frac{45}{4}-4z=-21$$

Tāpiri -\frac{65z}{16} ki te -4z.

$$-\frac{129}{16}z+\frac{45}{4}=-21$$

Me tango \frac{45}{4} mai i ngā taha e rua o te whārite.

$$-\frac{129}{16}z=-\frac{129}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{129}{16}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$z=4$$

Whakaurua te 4 mō z ki y=\frac{13}{16}z-\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{13}{16}\times 4-\frac{9}{4}$$

Whakareatia \frac{13}{16} ki te 4.

$$y=\frac{13-9}{4}$$

Tāpiri -\frac{9}{4} ki te \frac{13}{4} 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.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,z=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(6-5y+2z\right)=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6-5y+2z.

$$12-10y+4z=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 16+2y-3z.

$$12-10y+4z=48+6y-9z$$

Tangohia te 6y mai i ngā taha e rua.

$$12-10y+4z-6y=48-9z$$

Pahekotia te -10y me -6y, ka -16y.

$$12-16y+4z=48-9z$$

Me tāpiri te 9z ki ngā taha e rua.

$$12-16y+4z+9z=48$$

Pahekotia te 4z me 9z, ka 13z.

$$12-16y+13z=48$$

Tangohia te 12 mai i ngā taha e rua.

$$-16y+13z=48-12$$

Tangohia te 12 i te 48, ka 36.

$$-16y+13z=36$$

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.

$$4\left(6-5y+2z\right)=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 6-5y+2z.

$$24-20y+8z=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1-5y+4z.

$$24-20y+8z=3-15y+12z$$

Me tāpiri te 15y ki ngā taha e rua.

$$24-20y+8z+15y=3+12z$$

Pahekotia te -20y me 15y, ka -5y.

$$24-5y+8z=3+12z$$

Tangohia te 12z mai i ngā taha e rua.

$$24-5y+8z-12z=3$$

Pahekotia te 8z me -12z, ka -4z.

$$24-5y-4z=3$$

Tangohia te 24 mai i ngā taha e rua.

$$-5y-4z=3-24$$

Tangohia te 24 i te 3, ka -21.

$$-5y-4z=-21$$

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

$$-16y+13z=36,-5y-4z=-21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}36\\-21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right))\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right))\left(\begin{matrix}36\\-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right))\left(\begin{matrix}36\\-21\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\z\end{matrix}\right)=inverse(\left(\begin{matrix}-16&13\\-5&-4\end{matrix}\right))\left(\begin{matrix}36\\-21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-16\left(-4\right)-13\left(-5\right)}&-\frac{13}{-16\left(-4\right)-13\left(-5\right)}\\-\frac{-5}{-16\left(-4\right)-13\left(-5\right)}&-\frac{16}{-16\left(-4\right)-13\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}36\\-21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{129}&-\frac{13}{129}\\\frac{5}{129}&-\frac{16}{129}\end{matrix}\right)\left(\begin{matrix}36\\-21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{129}\times 36-\frac{13}{129}\left(-21\right)\\\frac{5}{129}\times 36-\frac{16}{129}\left(-21\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\z\end{matrix}\right)=\left(\begin{matrix}1\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me z.

$$y=1,z=4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(6-5y+2z\right)=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 6-5y+2z.

$$12-10y+4z=3\left(16+2y-3z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 16+2y-3z.

$$12-10y+4z=48+6y-9z$$

Tangohia te 6y mai i ngā taha e rua.

$$12-10y+4z-6y=48-9z$$

Pahekotia te -10y me -6y, ka -16y.

$$12-16y+4z=48-9z$$

Me tāpiri te 9z ki ngā taha e rua.

$$12-16y+4z+9z=48$$

Pahekotia te 4z me 9z, ka 13z.

$$12-16y+13z=48$$

Tangohia te 12 mai i ngā taha e rua.

$$-16y+13z=48-12$$

Tangohia te 12 i te 48, ka 36.

$$-16y+13z=36$$

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.

$$4\left(6-5y+2z\right)=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 6-5y+2z.

$$24-20y+8z=3\left(1-5y+4z\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 1-5y+4z.

$$24-20y+8z=3-15y+12z$$

Me tāpiri te 15y ki ngā taha e rua.

$$24-20y+8z+15y=3+12z$$

Pahekotia te -20y me 15y, ka -5y.

$$24-5y+8z=3+12z$$

Tangohia te 12z mai i ngā taha e rua.

$$24-5y+8z-12z=3$$

Pahekotia te 8z me -12z, ka -4z.

$$24-5y-4z=3$$

Tangohia te 24 mai i ngā taha e rua.

$$-5y-4z=3-24$$

Tangohia te 24 i te 3, ka -21.

$$-5y-4z=-21$$

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.

$$-16y+13z=36,-5y-4z=-21$$

Kia ōrite ai a -16y me -5y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -16.

$$-5\left(-16\right)y-5\times 13z=-5\times 36,-16\left(-5\right)y-16\left(-4\right)z=-16\left(-21\right)$$

Whakarūnātia.

$$80y-65z=-180,80y+64z=336$$

Me tango 80y+64z=336 mai i 80y-65z=-180 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$80y-80y-65z-64z=-180-336$$

Tāpiri 80y ki te -80y. Ka whakakore atu ngā kupu 80y me -80y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-65z-64z=-180-336$$

Tāpiri -65z ki te -64z.

$$-129z=-180-336$$

Tāpiri -180 ki te -336.

$$-129z=-516$$

Whakawehea ngā taha e rua ki te -129.

$$z=4$$

Whakaurua te 4 mō z ki -5y-4z=-21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-5y-4\times 4=-21$$

Whakareatia -4 ki te 4.

$$-5y-16=-21$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$-5y=-5$$

Whakawehea ngā taha e rua ki te -5.

$$y=1$$

Kua oti te pūnaha te whakatau.

$$y=1,z=4$$