Whakaoti i te {l}{3x=-4y}{5x-6y=38}

$$\left\{ \begin{array} { l } { 3 x = - 4 y } \\ { 5 x - 6 y = 38 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=-3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$3x+4y=0$$

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.

$$3x+4y=0,5x-6y=38$$

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

$$3x+4y=0$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$3x=-4y$$

Whakawehea ngā taha e rua ki te 3.

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

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

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

Whakakapia te -\frac{4y}{3} mō te x ki tērā atu whārite, 5x-6y=38.

$$5\left(-\frac{4}{3}\right)y-6y=38$$

Whakareatia 5 ki te -\frac{4y}{3}.

$$-\frac{20}{3}y-6y=38$$

Tāpiri -\frac{20y}{3} ki te -6y.

$$-\frac{38}{3}y=38$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{38}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-3$$

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

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

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

$$x=4$$

Kua oti te pūnaha te whakatau.

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

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3x+4y=0$$

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

$$3x+4y=0,5x-6y=38$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&4\\5&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\38\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}3&4\\5&-6\end{matrix}\right))\left(\begin{matrix}3&4\\5&-6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&-6\end{matrix}\right))\left(\begin{matrix}0\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&-6\end{matrix}\right))\left(\begin{matrix}0\\38\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\5&-6\end{matrix}\right))\left(\begin{matrix}0\\38\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{6}{3\left(-6\right)-4\times 5}&-\frac{4}{3\left(-6\right)-4\times 5}\\-\frac{5}{3\left(-6\right)-4\times 5}&\frac{3}{3\left(-6\right)-4\times 5}\end{matrix}\right)\left(\begin{matrix}0\\38\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{19}&\frac{2}{19}\\\frac{5}{38}&-\frac{3}{38}\end{matrix}\right)\left(\begin{matrix}0\\38\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{19}\times 38\\-\frac{3}{38}\times 38\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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

$$3x+4y=0$$

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.

$$3x+4y=0,5x-6y=38$$

Kia ōrite ai a 3x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$5\times 3x+5\times 4y=0,3\times 5x+3\left(-6\right)y=3\times 38$$

Whakarūnātia.

$$15x+20y=0,15x-18y=114$$

Me tango 15x-18y=114 mai i 15x+20y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+20y+18y=-114$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y+18y=-114$$

Tāpiri 20y ki te 18y.

$$38y=-114$$

Whakawehea ngā taha e rua ki te 38.

$$y=-3$$

Whakaurua te -3 mō y ki 5x-6y=38. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-6\left(-3\right)=38$$

Whakareatia -6 ki te -3.

$$5x+18=38$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Kua oti te pūnaha te whakatau.

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