Whakaoti i te {l}{x+1/y+1=1/3}{x-1/y-1=1/4}

$$\left. \begin{array}{l}{ \frac { x + 1 } { y + 1 } = \frac { 1 } { 3 } }\\{ \frac { x - 1 } { y - 1 } = \frac { 1 } { 4 } }\end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=17$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+1,3.

$$3\left(x+1\right)=y+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

$$3x+3=y+1$$

Tangohia te y mai i ngā taha e rua.

$$3x+3-y=1$$

Tangohia te 3 mai i ngā taha e rua.

$$3x-y=1-3$$

Tangohia te 3 i te 1, ka -2.

$$3x-y=-2$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,4.

$$4\left(x-1\right)=y-1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-1.

$$4x-4=y-1$$

Tangohia te y mai i ngā taha e rua.

$$4x-4-y=-1$$

Me tāpiri te 4 ki ngā taha e rua.

$$4x-y=-1+4$$

Tāpirihia te -1 ki te 4, ka 3.

$$4x-y=3$$

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.

$$3x-y=-2,4x-y=3$$

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

$$3x-y=-2$$

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

$$3x=y-2$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{1}{3}y-\frac{2}{3}$$

Whakakapia te \frac{-2+y}{3} mō te x ki tērā atu whārite, 4x-y=3.

$$4\left(\frac{1}{3}y-\frac{2}{3}\right)-y=3$$

Whakareatia 4 ki te \frac{-2+y}{3}.

$$\frac{4}{3}y-\frac{8}{3}-y=3$$

Tāpiri \frac{4y}{3} ki te -y.

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

Me tāpiri \frac{8}{3} ki ngā taha e rua o te whārite.

$$\frac{1}{3}y=\frac{17}{3}$$

Me whakarea ngā taha e rua ki te 3.

$$y=17$$

Whakaurua te 17 mō y ki x=\frac{1}{3}y-\frac{2}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times 17-\frac{2}{3}$$

Whakareatia \frac{1}{3} ki te 17.

$$x=\frac{17-2}{3}$$

Tāpiri -\frac{2}{3} ki te \frac{17}{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.

$$x=5,y=17$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+1,3.

$$3\left(x+1\right)=y+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

$$3x+3=y+1$$

Tangohia te y mai i ngā taha e rua.

$$3x+3-y=1$$

Tangohia te 3 mai i ngā taha e rua.

$$3x-y=1-3$$

Tangohia te 3 i te 1, ka -2.

$$3x-y=-2$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,4.

$$4\left(x-1\right)=y-1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-1.

$$4x-4=y-1$$

Tangohia te y mai i ngā taha e rua.

$$4x-4-y=-1$$

Me tāpiri te 4 ki ngā taha e rua.

$$4x-y=-1+4$$

Tāpirihia te -1 ki te 4, ka 3.

$$4x-y=3$$

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

$$3x-y=-2,4x-y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\4&-1\end{matrix}\right))\left(\begin{matrix}-2\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-4\right)}&-\frac{-1}{3\left(-1\right)-\left(-4\right)}\\-\frac{4}{3\left(-1\right)-\left(-4\right)}&\frac{3}{3\left(-1\right)-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}-2\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\-4&3\end{matrix}\right)\left(\begin{matrix}-2\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\left(-2\right)+3\\-4\left(-2\right)+3\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\17\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=17$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tē taea kia ōrite te tāupe y ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(y+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y+1,3.

$$3\left(x+1\right)=y+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+1.

$$3x+3=y+1$$

Tangohia te y mai i ngā taha e rua.

$$3x+3-y=1$$

Tangohia te 3 mai i ngā taha e rua.

$$3x-y=1-3$$

Tangohia te 3 i te 1, ka -2.

$$3x-y=-2$$

Whakaarohia te whārite tuarua. Tē taea kia ōrite te tāupe y ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(y-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o y-1,4.

$$4\left(x-1\right)=y-1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-1.

$$4x-4=y-1$$

Tangohia te y mai i ngā taha e rua.

$$4x-4-y=-1$$

Me tāpiri te 4 ki ngā taha e rua.

$$4x-y=-1+4$$

Tāpirihia te -1 ki te 4, ka 3.

$$4x-y=3$$

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.

$$3x-y=-2,4x-y=3$$

Me tango 4x-y=3 mai i 3x-y=-2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-4x-y+y=-2-3$$

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

$$3x-4x=-2-3$$

Tāpiri 3x ki te -4x.

$$-x=-2-3$$

Tāpiri -2 ki te -3.

$$-x=-5$$

Whakawehea ngā taha e rua ki te -1.

$$x=5$$

Whakaurua te 5 mō x ki 4x-y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4\times 5-y=3$$

Whakareatia 4 ki te 5.

$$20-y=3$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-y=-17$$

Whakawehea ngā taha e rua ki te -1.

$$y=17$$

Kua oti te pūnaha te whakatau.

$$x=5,y=17$$