Whakaoti i te {l}{2x-15=3(y+2)}{7(x-4)=-1-5y}

$$\left\{ \begin{array} { l } { 2 x - 15 = 3 ( y + 2 ) } \\ { 7 ( x - 4 ) = - 1 - 5 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x-15=3y+6$$

Tangohia te 3y mai i ngā taha e rua.

$$2x-15-3y=6$$

Me tāpiri te 15 ki ngā taha e rua.

$$2x-3y=6+15$$

Tāpirihia te 6 ki te 15, ka 21.

$$2x-3y=21$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$7x-28=-1-5y$$

Me tāpiri te 5y ki ngā taha e rua.

$$7x-28+5y=-1$$

Me tāpiri te 28 ki ngā taha e rua.

$$7x+5y=-1+28$$

Tāpirihia te -1 ki te 28, ka 27.

$$7x+5y=27$$

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.

$$2x-3y=21,7x+5y=27$$

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

$$2x-3y=21$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$2x=3y+21$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+21\right)$$

Whakareatia \frac{1}{2} ki te 21+3y.

$$x=\frac{3}{2}y+\frac{21}{2}$$

Whakakapia te \frac{21+3y}{2} mō te x ki tērā atu whārite, 7x+5y=27.

$$7\left(\frac{3}{2}y+\frac{21}{2}\right)+5y=27$$

Whakareatia 7 ki te \frac{21+3y}{2}.

$$\frac{21}{2}y+\frac{147}{2}+5y=27$$

Tāpiri \frac{21y}{2} ki te 5y.

$$\frac{31}{2}y+\frac{147}{2}=27$$

Me tango \frac{147}{2} mai i ngā taha e rua o te whārite.

$$\frac{31}{2}y=-\frac{93}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{31}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

Whakaurua te -3 mō y ki x=\frac{3}{2}y+\frac{21}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\left(-3\right)+\frac{21}{2}$$

Whakareatia \frac{3}{2} ki te -3.

$$x=\frac{-9+21}{2}$$

Tāpiri \frac{21}{2} ki te -\frac{9}{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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x-15=3y+6$$

Tangohia te 3y mai i ngā taha e rua.

$$2x-15-3y=6$$

Me tāpiri te 15 ki ngā taha e rua.

$$2x-3y=6+15$$

Tāpirihia te 6 ki te 15, ka 21.

$$2x-3y=21$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$7x-28=-1-5y$$

Me tāpiri te 5y ki ngā taha e rua.

$$7x-28+5y=-1$$

Me tāpiri te 28 ki ngā taha e rua.

$$7x+5y=-1+28$$

Tāpirihia te -1 ki te 28, ka 27.

$$7x+5y=27$$

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

$$2x-3y=21,7x+5y=27$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\27\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-3\\7&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\7&5\end{matrix}\right))\left(\begin{matrix}2&-3\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&5\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&5\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\7&5\end{matrix}\right))\left(\begin{matrix}21\\27\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-\left(-3\times 7\right)}&-\frac{-3}{2\times 5-\left(-3\times 7\right)}\\-\frac{7}{2\times 5-\left(-3\times 7\right)}&\frac{2}{2\times 5-\left(-3\times 7\right)}\end{matrix}\right)\left(\begin{matrix}21\\27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}&\frac{3}{31}\\-\frac{7}{31}&\frac{2}{31}\end{matrix}\right)\left(\begin{matrix}21\\27\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{31}\times 21+\frac{3}{31}\times 27\\-\frac{7}{31}\times 21+\frac{2}{31}\times 27\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-3$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te y+2.

$$2x-15=3y+6$$

Tangohia te 3y mai i ngā taha e rua.

$$2x-15-3y=6$$

Me tāpiri te 15 ki ngā taha e rua.

$$2x-3y=6+15$$

Tāpirihia te 6 ki te 15, ka 21.

$$2x-3y=21$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-4.

$$7x-28=-1-5y$$

Me tāpiri te 5y ki ngā taha e rua.

$$7x-28+5y=-1$$

Me tāpiri te 28 ki ngā taha e rua.

$$7x+5y=-1+28$$

Tāpirihia te -1 ki te 28, ka 27.

$$7x+5y=27$$

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.

$$2x-3y=21,7x+5y=27$$

Kia ōrite ai a 2x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$7\times 2x+7\left(-3\right)y=7\times 21,2\times 7x+2\times 5y=2\times 27$$

Whakarūnātia.

$$14x-21y=147,14x+10y=54$$

Me tango 14x+10y=54 mai i 14x-21y=147 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x-21y-10y=147-54$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-21y-10y=147-54$$

Tāpiri -21y ki te -10y.

$$-31y=147-54$$

Tāpiri 147 ki te -54.

$$-31y=93$$

Whakawehea ngā taha e rua ki te -31.

$$y=-3$$

Whakaurua te -3 mō y ki 7x+5y=27. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+5\left(-3\right)=27$$

Whakareatia 5 ki te -3.

$$7x-15=27$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$7x=42$$

Whakawehea ngā taha e rua ki te 7.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-3$$