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

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

Whakaoti mō x, y

$$x=5$$

$$y=7$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$xy+5x-2y-10=\left(x-1\right)\left(y+2\right)$$

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

$$xy+5x-2y-10=xy+2x-y-2$$

Tangohia te xy mai i ngā taha e rua.

$$xy+5x-2y-10-xy=2x-y-2$$

Pahekotia te xy me -xy, ka 0.

$$5x-2y-10=2x-y-2$$

Tangohia te 2x mai i ngā taha e rua.

$$5x-2y-10-2x=-y-2$$

Pahekotia te 5x me -2x, ka 3x.

$$3x-2y-10=-y-2$$

Me tāpiri te y ki ngā taha e rua.

$$3x-2y-10+y=-2$$

Pahekotia te -2y me y, ka -y.

$$3x-y-10=-2$$

Me tāpiri te 10 ki ngā taha e rua.

$$3x-y=-2+10$$

Tāpirihia te -2 ki te 10, ka 8.

$$3x-y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te y-3 ki te x+4.

$$yx+4y-3x-12=\left(x+7\right)\left(y-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te y-4.

$$yx+4y-3x-12=xy-4x+7y-28$$

Tangohia te xy mai i ngā taha e rua.

$$yx+4y-3x-12-xy=-4x+7y-28$$

Pahekotia te yx me -xy, ka 0.

$$4y-3x-12=-4x+7y-28$$

Me tāpiri te 4x ki ngā taha e rua.

$$4y-3x-12+4x=7y-28$$

Pahekotia te -3x me 4x, ka x.

$$4y+x-12=7y-28$$

Tangohia te 7y mai i ngā taha e rua.

$$4y+x-12-7y=-28$$

Pahekotia te 4y me -7y, ka -3y.

$$-3y+x-12=-28$$

Me tāpiri te 12 ki ngā taha e rua.

$$-3y+x=-28+12$$

Tāpirihia te -28 ki te 12, ka -16.

$$-3y+x=-16$$

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.

$$3x-y=8,x-3y=-16$$

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

$$3x-y=8$$

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

$$3x=y+8$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(y+8\right)$$

Whakareatia \frac{1}{3} ki te y+8.

$$x=\frac{1}{3}y+\frac{8}{3}$$

Whakakapia te \frac{8+y}{3} mō te x ki tērā atu whārite, x-3y=-16.

$$\frac{1}{3}y+\frac{8}{3}-3y=-16$$

Tāpiri \frac{y}{3} ki te -3y.

$$-\frac{8}{3}y+\frac{8}{3}=-16$$

Me tango \frac{8}{3} mai i ngā taha e rua o te whārite.

$$-\frac{8}{3}y=-\frac{56}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{3}, 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}{3}y+\frac{8}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 7+\frac{8}{3}$$

Whakareatia \frac{1}{3} ki te 7.

$$x=\frac{7+8}{3}$$

Tāpiri \frac{8}{3} ki te \frac{7}{3} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$xy+5x-2y-10=\left(x-1\right)\left(y+2\right)$$

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

$$xy+5x-2y-10=xy+2x-y-2$$

Tangohia te xy mai i ngā taha e rua.

$$xy+5x-2y-10-xy=2x-y-2$$

Pahekotia te xy me -xy, ka 0.

$$5x-2y-10=2x-y-2$$

Tangohia te 2x mai i ngā taha e rua.

$$5x-2y-10-2x=-y-2$$

Pahekotia te 5x me -2x, ka 3x.

$$3x-2y-10=-y-2$$

Me tāpiri te y ki ngā taha e rua.

$$3x-2y-10+y=-2$$

Pahekotia te -2y me y, ka -y.

$$3x-y-10=-2$$

Me tāpiri te 10 ki ngā taha e rua.

$$3x-y=-2+10$$

Tāpirihia te -2 ki te 10, ka 8.

$$3x-y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te y-3 ki te x+4.

$$yx+4y-3x-12=\left(x+7\right)\left(y-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te y-4.

$$yx+4y-3x-12=xy-4x+7y-28$$

Tangohia te xy mai i ngā taha e rua.

$$yx+4y-3x-12-xy=-4x+7y-28$$

Pahekotia te yx me -xy, ka 0.

$$4y-3x-12=-4x+7y-28$$

Me tāpiri te 4x ki ngā taha e rua.

$$4y-3x-12+4x=7y-28$$

Pahekotia te -3x me 4x, ka x.

$$4y+x-12=7y-28$$

Tangohia te 7y mai i ngā taha e rua.

$$4y+x-12-7y=-28$$

Pahekotia te 4y me -7y, ka -3y.

$$-3y+x-12=-28$$

Me tāpiri te 12 ki ngā taha e rua.

$$-3y+x=-28+12$$

Tāpirihia te -28 ki te 12, ka -16.

$$-3y+x=-16$$

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

$$3x-y=8,x-3y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right))\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\1&-3\end{matrix}\right))\left(\begin{matrix}8\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-\left(-1\right)}&-\frac{-1}{3\left(-3\right)-\left(-1\right)}\\-\frac{1}{3\left(-3\right)-\left(-1\right)}&\frac{3}{3\left(-3\right)-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}8\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&-\frac{1}{8}\\\frac{1}{8}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}8\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 8-\frac{1}{8}\left(-16\right)\\\frac{1}{8}\times 8-\frac{3}{8}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=7$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$xy+5x-2y-10=\left(x-1\right)\left(y+2\right)$$

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

$$xy+5x-2y-10=xy+2x-y-2$$

Tangohia te xy mai i ngā taha e rua.

$$xy+5x-2y-10-xy=2x-y-2$$

Pahekotia te xy me -xy, ka 0.

$$5x-2y-10=2x-y-2$$

Tangohia te 2x mai i ngā taha e rua.

$$5x-2y-10-2x=-y-2$$

Pahekotia te 5x me -2x, ka 3x.

$$3x-2y-10=-y-2$$

Me tāpiri te y ki ngā taha e rua.

$$3x-2y-10+y=-2$$

Pahekotia te -2y me y, ka -y.

$$3x-y-10=-2$$

Me tāpiri te 10 ki ngā taha e rua.

$$3x-y=-2+10$$

Tāpirihia te -2 ki te 10, ka 8.

$$3x-y=8$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te y-3 ki te x+4.

$$yx+4y-3x-12=\left(x+7\right)\left(y-4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+7 ki te y-4.

$$yx+4y-3x-12=xy-4x+7y-28$$

Tangohia te xy mai i ngā taha e rua.

$$yx+4y-3x-12-xy=-4x+7y-28$$

Pahekotia te yx me -xy, ka 0.

$$4y-3x-12=-4x+7y-28$$

Me tāpiri te 4x ki ngā taha e rua.

$$4y-3x-12+4x=7y-28$$

Pahekotia te -3x me 4x, ka x.

$$4y+x-12=7y-28$$

Tangohia te 7y mai i ngā taha e rua.

$$4y+x-12-7y=-28$$

Pahekotia te 4y me -7y, ka -3y.

$$-3y+x-12=-28$$

Me tāpiri te 12 ki ngā taha e rua.

$$-3y+x=-28+12$$

Tāpirihia te -28 ki te 12, ka -16.

$$-3y+x=-16$$

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.

$$3x-y=8,x-3y=-16$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x-y=8,3x+3\left(-3\right)y=3\left(-16\right)$$

Whakarūnātia.

$$3x-y=8,3x-9y=-48$$

Me tango 3x-9y=-48 mai i 3x-y=8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x-y+9y=8+48$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-y+9y=8+48$$

Tāpiri -y ki te 9y.

$$8y=8+48$$

Tāpiri 8 ki te 48.

$$8y=56$$

Whakawehea ngā taha e rua ki te 8.

$$y=7$$

Whakaurua te 7 mō y ki x-3y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\times 7=-16$$

Whakareatia -3 ki te 7.

$$x-21=-16$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$