Whakaoti i te {l}{x/3+y/4=2/2-6/6}{frac{2x+y}{5}-frac{y-2}{2}=frac{x+y-3}{4}-frac{y-x-1}{10}}

$$\left\{ \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 4 } = \frac { 2 } { 2 } - \frac { 6 } { 6 } } \\ { \frac { 2 x + y } { 5 } - \frac { y - 2 } { 2 } = \frac { x + y - 3 } { 4 } - \frac { y - x - 1 } { 10 } } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{33}{13} = -2\frac{7}{13} \approx -2.538461538$$

$$y = \frac{44}{13} = 3\frac{5}{13} \approx 3.384615385$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,2,6.

$$4x+3y=6\times 2-2\times 6$$

Mahia ngā whakarea.

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

Tangohia te 12 i te 12, ka 0.

$$4x+3y=0$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2,4,10.

$$4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x+y.

$$8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -10 ki te y-2.

$$8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Pahekotia te 4y me -10y, ka -6y.

$$8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2y+2x+2$$

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

$$8x-6y+20=5x+3y-15+2x+2$$

Pahekotia te 5x me 2x, ka 7x.

$$8x-6y+20=7x+3y-15+2$$

Tāpirihia te -15 ki te 2, ka -13.

$$8x-6y+20=7x+3y-13$$

Tangohia te 7x mai i ngā taha e rua.

$$8x-6y+20-7x=3y-13$$

Pahekotia te 8x me -7x, ka x.

$$x-6y+20=3y-13$$

Tangohia te 3y mai i ngā taha e rua.

$$x-6y+20-3y=-13$$

Pahekotia te -6y me -3y, ka -9y.

$$x-9y+20=-13$$

Tangohia te 20 mai i ngā taha e rua.

$$x-9y=-13-20$$

Tangohia te 20 i te -13, ka -33.

$$x-9y=-33$$

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.

$$4x+3y=0,x-9y=-33$$

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

$$4x+3y=0$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$4x=-3y$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

Whakakapia te -\frac{3y}{4} mō te x ki tērā atu whārite, x-9y=-33.

$$-\frac{3}{4}y-9y=-33$$

Tāpiri -\frac{3y}{4} ki te -9y.

$$-\frac{39}{4}y=-33$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{39}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{44}{13}$$

Whakaurua te \frac{44}{13} mō y ki x=-\frac{3}{4}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times \frac{44}{13}$$

Whakareatia -\frac{3}{4} ki te \frac{44}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{33}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{33}{13},y=\frac{44}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,2,6.

$$4x+3y=6\times 2-2\times 6$$

Mahia ngā whakarea.

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

Tangohia te 12 i te 12, ka 0.

$$4x+3y=0$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2,4,10.

$$4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x+y.

$$8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -10 ki te y-2.

$$8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Pahekotia te 4y me -10y, ka -6y.

$$8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2y+2x+2$$

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

$$8x-6y+20=5x+3y-15+2x+2$$

Pahekotia te 5x me 2x, ka 7x.

$$8x-6y+20=7x+3y-15+2$$

Tāpirihia te -15 ki te 2, ka -13.

$$8x-6y+20=7x+3y-13$$

Tangohia te 7x mai i ngā taha e rua.

$$8x-6y+20-7x=3y-13$$

Pahekotia te 8x me -7x, ka x.

$$x-6y+20=3y-13$$

Tangohia te 3y mai i ngā taha e rua.

$$x-6y+20-3y=-13$$

Pahekotia te -6y me -3y, ka -9y.

$$x-9y+20=-13$$

Tangohia te 20 mai i ngā taha e rua.

$$x-9y=-13-20$$

Tangohia te 20 i te -13, ka -33.

$$x-9y=-33$$

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

$$4x+3y=0,x-9y=-33$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-33\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\1&-9\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}4&3\\1&-9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\1&-9\end{matrix}\right))\left(\begin{matrix}0\\-33\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{4\left(-9\right)-3}&-\frac{3}{4\left(-9\right)-3}\\-\frac{1}{4\left(-9\right)-3}&\frac{4}{4\left(-9\right)-3}\end{matrix}\right)\left(\begin{matrix}0\\-33\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{13}&\frac{1}{13}\\\frac{1}{39}&-\frac{4}{39}\end{matrix}\right)\left(\begin{matrix}0\\-33\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13}\left(-33\right)\\-\frac{4}{39}\left(-33\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{33}{13}\\\frac{44}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{33}{13},y=\frac{44}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,2,6.

$$4x+3y=6\times 2-2\times 6$$

Mahia ngā whakarea.

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

Tangohia te 12 i te 12, ka 0.

$$4x+3y=0$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,2,4,10.

$$4\left(2x+y\right)-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 2x+y.

$$8x+4y-10\left(y-2\right)=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -10 ki te y-2.

$$8x+4y-10y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

Pahekotia te 4y me -10y, ka -6y.

$$8x-6y+20=5\left(x+y-3\right)-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2\left(y-x-1\right)$$

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

$$8x-6y+20=5x+5y-15-2y+2x+2$$

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

$$8x-6y+20=5x+3y-15+2x+2$$

Pahekotia te 5x me 2x, ka 7x.

$$8x-6y+20=7x+3y-15+2$$

Tāpirihia te -15 ki te 2, ka -13.

$$8x-6y+20=7x+3y-13$$

Tangohia te 7x mai i ngā taha e rua.

$$8x-6y+20-7x=3y-13$$

Pahekotia te 8x me -7x, ka x.

$$x-6y+20=3y-13$$

Tangohia te 3y mai i ngā taha e rua.

$$x-6y+20-3y=-13$$

Pahekotia te -6y me -3y, ka -9y.

$$x-9y+20=-13$$

Tangohia te 20 mai i ngā taha e rua.

$$x-9y=-13-20$$

Tangohia te 20 i te -13, ka -33.

$$x-9y=-33$$

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.

$$4x+3y=0,x-9y=-33$$

Kia ōrite ai a 4x 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 4.

$$4x+3y=0,4x+4\left(-9\right)y=4\left(-33\right)$$

Whakarūnātia.

$$4x+3y=0,4x-36y=-132$$

Me tango 4x-36y=-132 mai i 4x+3y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4x-4x+3y+36y=132$$

Tāpiri 4x ki te -4x. Ka whakakore atu ngā kupu 4x me -4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+36y=132$$

Tāpiri 3y ki te 36y.

$$39y=132$$

Whakawehea ngā taha e rua ki te 39.

$$y=\frac{44}{13}$$

Whakaurua te \frac{44}{13} mō y ki x-9y=-33. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-9\times \frac{44}{13}=-33$$

Whakareatia -9 ki te \frac{44}{13}.

$$x-\frac{396}{13}=-33$$

Me tāpiri \frac{396}{13} ki ngā taha e rua o te whārite.

$$x=-\frac{33}{13}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{33}{13},y=\frac{44}{13}$$