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

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

Whakaoti mō x, y

$$x=-9$$

$$y=-12$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=0$$

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-x=-3$$

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.

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

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

$$4x-3y=0$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$4x=3y$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\times 3y$$

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+y=-3.

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

Tāpiri -\frac{3y}{4} ki te y.

$$\frac{1}{4}y=-3$$

Me whakarea ngā taha e rua ki te 4.

$$y=-12$$

Whakaurua te -12 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.

$$x=\frac{3}{4}\left(-12\right)$$

Whakareatia \frac{3}{4} ki te -12.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=-12$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=0$$

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-x=-3$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-9\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-9,y=-12$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3y mai i ngā taha e rua.

$$4x-3y=0$$

Whakaarohia te whārite tuarua. Tangohia te x mai i ngā taha e rua.

$$y+3-x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$y-x=-3$$

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.

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

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.

$$-4x-\left(-3y\right)=0,4\left(-1\right)x+4y=4\left(-3\right)$$

Whakarūnātia.

$$-4x+3y=0,-4x+4y=-12$$

Me tango -4x+4y=-12 mai i -4x+3y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

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

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.

$$3y-4y=12$$

Tāpiri 3y ki te -4y.

$$-y=12$$

Whakawehea ngā taha e rua ki te -1.

$$y=-12$$

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

$$-x-12=-3$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$-x=9$$

Whakawehea ngā taha e rua ki te -1.

$$x=-9$$

Kua oti te pūnaha te whakatau.

$$x=-9,y=-12$$