Whakaoti i te {l}{5(x+2)=4y}{3(y-4)=6x}

$$\left. \begin{array} { l } { 5 ( x + 2 ) = 4 y } \\ { 3 ( y - 4 ) = 6 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$5x+10=4y$$

Tangohia te 4y mai i ngā taha e rua.

$$5x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x-4y=-10$$

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

$$3y-12=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$3y-12-6x=0$$

Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-6x=12$$

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.

$$5x-4y=-10,-6x+3y=12$$

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

$$5x-4y=-10$$

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

$$5x=4y-10$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(4y-10\right)$$

Whakareatia \frac{1}{5} ki te 4y-10.

$$x=\frac{4}{5}y-2$$

Whakakapia te \frac{4y}{5}-2 mō te x ki tērā atu whārite, -6x+3y=12.

$$-6\left(\frac{4}{5}y-2\right)+3y=12$$

Whakareatia -6 ki te \frac{4y}{5}-2.

$$-\frac{24}{5}y+12+3y=12$$

Tāpiri -\frac{24y}{5} ki te 3y.

$$-\frac{9}{5}y+12=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-\frac{9}{5}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{4}{5}y-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$5x+10=4y$$

Tangohia te 4y mai i ngā taha e rua.

$$5x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x-4y=-10$$

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

$$3y-12=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$3y-12-6x=0$$

Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-6x=12$$

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

$$5x-4y=-10,-6x+3y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-4\\-6&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right))\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-4\\-6&3\end{matrix}\right))\left(\begin{matrix}-10\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-\left(-4\left(-6\right)\right)}&-\frac{-4}{5\times 3-\left(-4\left(-6\right)\right)}\\-\frac{-6}{5\times 3-\left(-4\left(-6\right)\right)}&\frac{5}{5\times 3-\left(-4\left(-6\right)\right)}\end{matrix}\right)\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}&-\frac{4}{9}\\-\frac{2}{3}&-\frac{5}{9}\end{matrix}\right)\left(\begin{matrix}-10\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3}\left(-10\right)-\frac{4}{9}\times 12\\-\frac{2}{3}\left(-10\right)-\frac{5}{9}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-2\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$5x+10=4y$$

Tangohia te 4y mai i ngā taha e rua.

$$5x+10-4y=0$$

Tangohia te 10 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x-4y=-10$$

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

$$3y-12=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$3y-12-6x=0$$

Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3y-6x=12$$

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.

$$5x-4y=-10,-6x+3y=12$$

Kia ōrite ai a 5x me -6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-6\times 5x-6\left(-4\right)y=-6\left(-10\right),5\left(-6\right)x+5\times 3y=5\times 12$$

Whakarūnātia.

$$-30x+24y=60,-30x+15y=60$$

Me tango -30x+15y=60 mai i -30x+24y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-30x+30x+24y-15y=60-60$$

Tāpiri -30x ki te 30x. Ka whakakore atu ngā kupu -30x me 30x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$24y-15y=60-60$$

Tāpiri 24y ki te -15y.

$$9y=60-60$$

Tāpiri 60 ki te -60.

$$9y=0$$

Whakawehea ngā taha e rua ki te 9.

$$y=0$$

Whakaurua te 0 mō y ki -6x+3y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-6x=12$$

Whakawehea ngā taha e rua ki te -6.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=0$$