Whakaoti i te {l}{3(5x-2)-7(2y+3)=2}{2(3x-y)-23=3(4-9x)}

$$\left\{ \begin{array} { l } { 3 ( 5 x - 2 ) - 7 ( 2 y + 3 ) = 2 } \\ { 2 ( 3 x - y ) - 23 = 3 ( 4 - 9 x ) } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5x-2.

$$15x-6-7\left(2y+3\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+3.

$$15x-6-14y-21=2$$

Tangohia te 21 i te -6, ka -27.

$$15x-27-14y=2$$

Me tāpiri te 27 ki ngā taha e rua.

$$15x-14y=2+27$$

Tāpirihia te 2 ki te 27, ka 29.

$$15x-14y=29$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-y.

$$6x-2y-23=3\left(4-9x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 4-9x.

$$6x-2y-23=12-27x$$

Me tāpiri te 27x ki ngā taha e rua.

$$6x-2y-23+27x=12$$

Pahekotia te 6x me 27x, ka 33x.

$$33x-2y-23=12$$

Me tāpiri te 23 ki ngā taha e rua.

$$33x-2y=12+23$$

Tāpirihia te 12 ki te 23, ka 35.

$$33x-2y=35$$

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.

$$15x-14y=29,33x-2y=35$$

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

$$15x-14y=29$$

Me tāpiri 14y ki ngā taha e rua o te whārite.

$$15x=14y+29$$

Whakawehea ngā taha e rua ki te 15.

$$x=\frac{1}{15}\left(14y+29\right)$$

Whakareatia \frac{1}{15} ki te 14y+29.

$$x=\frac{14}{15}y+\frac{29}{15}$$

Whakakapia te \frac{14y+29}{15} mō te x ki tērā atu whārite, 33x-2y=35.

$$33\left(\frac{14}{15}y+\frac{29}{15}\right)-2y=35$$

Whakareatia 33 ki te \frac{14y+29}{15}.

$$\frac{154}{5}y+\frac{319}{5}-2y=35$$

Tāpiri \frac{154y}{5} ki te -2y.

$$\frac{144}{5}y+\frac{319}{5}=35$$

Me tango \frac{319}{5} mai i ngā taha e rua o te whārite.

$$\frac{144}{5}y=-\frac{144}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{144}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=\frac{14}{15}y+\frac{29}{15}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{14}{15}\left(-1\right)+\frac{29}{15}$$

Whakareatia \frac{14}{15} ki te -1.

$$x=\frac{-14+29}{15}$$

Tāpiri \frac{29}{15} ki te -\frac{14}{15} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5x-2.

$$15x-6-7\left(2y+3\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+3.

$$15x-6-14y-21=2$$

Tangohia te 21 i te -6, ka -27.

$$15x-27-14y=2$$

Me tāpiri te 27 ki ngā taha e rua.

$$15x-14y=2+27$$

Tāpirihia te 2 ki te 27, ka 29.

$$15x-14y=29$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-y.

$$6x-2y-23=3\left(4-9x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 4-9x.

$$6x-2y-23=12-27x$$

Me tāpiri te 27x ki ngā taha e rua.

$$6x-2y-23+27x=12$$

Pahekotia te 6x me 27x, ka 33x.

$$33x-2y-23=12$$

Me tāpiri te 23 ki ngā taha e rua.

$$33x-2y=12+23$$

Tāpirihia te 12 ki te 23, ka 35.

$$33x-2y=35$$

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

$$15x-14y=29,33x-2y=35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}29\\35\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}15&-14\\33&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right))\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right))\left(\begin{matrix}29\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right))\left(\begin{matrix}29\\35\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&-14\\33&-2\end{matrix}\right))\left(\begin{matrix}29\\35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{15\left(-2\right)-\left(-14\times 33\right)}&-\frac{-14}{15\left(-2\right)-\left(-14\times 33\right)}\\-\frac{33}{15\left(-2\right)-\left(-14\times 33\right)}&\frac{15}{15\left(-2\right)-\left(-14\times 33\right)}\end{matrix}\right)\left(\begin{matrix}29\\35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{216}&\frac{7}{216}\\-\frac{11}{144}&\frac{5}{144}\end{matrix}\right)\left(\begin{matrix}29\\35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{216}\times 29+\frac{7}{216}\times 35\\-\frac{11}{144}\times 29+\frac{5}{144}\times 35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 5x-2.

$$15x-6-7\left(2y+3\right)=2$$

Whakamahia te āhuatanga tohatoha hei whakarea te -7 ki te 2y+3.

$$15x-6-14y-21=2$$

Tangohia te 21 i te -6, ka -27.

$$15x-27-14y=2$$

Me tāpiri te 27 ki ngā taha e rua.

$$15x-14y=2+27$$

Tāpirihia te 2 ki te 27, ka 29.

$$15x-14y=29$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te 3x-y.

$$6x-2y-23=3\left(4-9x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 4-9x.

$$6x-2y-23=12-27x$$

Me tāpiri te 27x ki ngā taha e rua.

$$6x-2y-23+27x=12$$

Pahekotia te 6x me 27x, ka 33x.

$$33x-2y-23=12$$

Me tāpiri te 23 ki ngā taha e rua.

$$33x-2y=12+23$$

Tāpirihia te 12 ki te 23, ka 35.

$$33x-2y=35$$

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.

$$15x-14y=29,33x-2y=35$$

Kia ōrite ai a 15x me 33x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 33 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 15.

$$33\times 15x+33\left(-14\right)y=33\times 29,15\times 33x+15\left(-2\right)y=15\times 35$$

Whakarūnātia.

$$495x-462y=957,495x-30y=525$$

Me tango 495x-30y=525 mai i 495x-462y=957 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$495x-495x-462y+30y=957-525$$

Tāpiri 495x ki te -495x. Ka whakakore atu ngā kupu 495x me -495x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-462y+30y=957-525$$

Tāpiri -462y ki te 30y.

$$-432y=957-525$$

Tāpiri 957 ki te -525.

$$-432y=432$$

Whakawehea ngā taha e rua ki te -432.

$$y=-1$$

Whakaurua te -1 mō y ki 33x-2y=35. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$33x-2\left(-1\right)=35$$

Whakareatia -2 ki te -1.

$$33x+2=35$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$33x=33$$

Whakawehea ngā taha e rua ki te 33.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-1$$