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

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

Whakaoti mō y, x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

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.

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

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

$$y-4x=5$$

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

$$y=4x+5$$

Whakakapia te 4x+5 mō te y ki tērā atu whārite, -3y+4x=3.

$$-3\left(4x+5\right)+4x=3$$

Whakareatia -3 ki te 4x+5.

$$-12x-15+4x=3$$

Tāpiri -12x ki te 4x.

$$-8x-15=3$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$-8x=18$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{9}{4}$$

Whakaurua te -\frac{9}{4} mō x ki y=4x+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=4\left(-\frac{9}{4}\right)+5$$

Whakareatia 4 ki te -\frac{9}{4}.

$$y=-9+5$$

Tāpiri 5 ki te -9.

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=-\frac{9}{4}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&-\frac{1}{2}\\-\frac{3}{8}&-\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}5\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 5-\frac{1}{2}\times 3\\-\frac{3}{8}\times 5-\frac{1}{8}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-4\\-\frac{9}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-4,x=-\frac{9}{4}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 4x mai i ngā taha e rua.

$$y-4x=5$$

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.

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

Kia ōrite ai a y me -3y, 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 1.

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

Whakarūnātia.

$$-3y+12x=-15,-3y+4x=3$$

Me tango -3y+4x=3 mai i -3y+12x=-15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-3y+3y+12x-4x=-15-3$$

Tāpiri -3y ki te 3y. Ka whakakore atu ngā kupu -3y me 3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12x-4x=-15-3$$

Tāpiri 12x ki te -4x.

$$8x=-15-3$$

Tāpiri -15 ki te -3.

$$8x=-18$$

Whakawehea ngā taha e rua ki te 8.

$$x=-\frac{9}{4}$$

Whakaurua te -\frac{9}{4} mō x ki -3y+4x=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-3y+4\left(-\frac{9}{4}\right)=3$$

Whakareatia 4 ki te -\frac{9}{4}.

$$-3y-9=3$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$-3y=12$$

Whakawehea ngā taha e rua ki te -3.

$$y=-4$$

Kua oti te pūnaha te whakatau.

$$y=-4,x=-\frac{9}{4}$$