Whakaoti i te {l}{3(x-1)-4(y-4)=0}{5(y-1)=3(x+5)}

$$\left\{ \begin{array} { l } { 3 ( x - 1 ) - 4 ( y - 4 ) = 0 } \\ { 5 ( y - 1 ) = 3 ( x + 5 ) } \end{array} \right.$$

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 3 ki te x-1.

$$3x-3-4\left(y-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-4.

$$3x-3-4y+16=0$$

Tāpirihia te -3 ki te 16, ka 13.

$$3x+13-4y=0$$

Tangohia te 13 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-4y=-13$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$5y-5=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-5-3x=15$$

Me tāpiri te 5 ki ngā taha e rua.

$$5y-3x=15+5$$

Tāpirihia te 15 ki te 5, ka 20.

$$5y-3x=20$$

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.

$$3x-4y=-13,-3x+5y=20$$

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

$$3x-4y=-13$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$3x=4y-13$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(4y-13\right)$$

Whakareatia \frac{1}{3} ki te 4y-13.

$$x=\frac{4}{3}y-\frac{13}{3}$$

Whakakapia te \frac{4y-13}{3} mō te x ki tērā atu whārite, -3x+5y=20.

$$-3\left(\frac{4}{3}y-\frac{13}{3}\right)+5y=20$$

Whakareatia -3 ki te \frac{4y-13}{3}.

$$-4y+13+5y=20$$

Tāpiri -4y ki te 5y.

$$y+13=20$$

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

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{4}{3}y-\frac{13}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{3}\times 7-\frac{13}{3}$$

Whakareatia \frac{4}{3} ki te 7.

$$x=\frac{28-13}{3}$$

Tāpiri -\frac{13}{3} ki te \frac{28}{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 3 ki te x-1.

$$3x-3-4\left(y-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-4.

$$3x-3-4y+16=0$$

Tāpirihia te -3 ki te 16, ka 13.

$$3x+13-4y=0$$

Tangohia te 13 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-4y=-13$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$5y-5=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-5-3x=15$$

Me tāpiri te 5 ki ngā taha e rua.

$$5y-3x=15+5$$

Tāpirihia te 15 ki te 5, ka 20.

$$5y-3x=20$$

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

$$3x-4y=-13,-3x+5y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-13\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-4\\-3&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right))\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right))\left(\begin{matrix}-13\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right))\left(\begin{matrix}-13\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-4\\-3&5\end{matrix}\right))\left(\begin{matrix}-13\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3\times 5-\left(-4\left(-3\right)\right)}&-\frac{-4}{3\times 5-\left(-4\left(-3\right)\right)}\\-\frac{-3}{3\times 5-\left(-4\left(-3\right)\right)}&\frac{3}{3\times 5-\left(-4\left(-3\right)\right)}\end{matrix}\right)\left(\begin{matrix}-13\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}&\frac{4}{3}\\1&1\end{matrix}\right)\left(\begin{matrix}-13\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{3}\left(-13\right)+\frac{4}{3}\times 20\\-13+20\end{matrix}\right)$$

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 3 ki te x-1.

$$3x-3-4\left(y-4\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te y-4.

$$3x-3-4y+16=0$$

Tāpirihia te -3 ki te 16, ka 13.

$$3x+13-4y=0$$

Tangohia te 13 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-4y=-13$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 5 ki te y-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+5.

$$5y-5=3x+15$$

Tangohia te 3x mai i ngā taha e rua.

$$5y-5-3x=15$$

Me tāpiri te 5 ki ngā taha e rua.

$$5y-3x=15+5$$

Tāpirihia te 15 ki te 5, ka 20.

$$5y-3x=20$$

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.

$$3x-4y=-13,-3x+5y=20$$

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

$$-3\times 3x-3\left(-4\right)y=-3\left(-13\right),3\left(-3\right)x+3\times 5y=3\times 20$$

Whakarūnātia.

$$-9x+12y=39,-9x+15y=60$$

Me tango -9x+15y=60 mai i -9x+12y=39 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-9x+9x+12y-15y=39-60$$

Tāpiri -9x ki te 9x. Ka whakakore atu ngā kupu -9x me 9x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-15y=39-60$$

Tāpiri 12y ki te -15y.

$$-3y=39-60$$

Tāpiri 39 ki te -60.

$$-3y=-21$$

Whakawehea ngā taha e rua ki te -3.

$$y=7$$

Whakaurua te 7 mō y ki -3x+5y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-3x+5\times 7=20$$

Whakareatia 5 ki te 7.

$$-3x+35=20$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$-3x=-15$$

Whakawehea ngā taha e rua ki te -3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=7$$