Whakaoti i te 4x-y=5text{and}5y-4x=7

$$4 x - y = 5 \text { and } 5 y - 4 x = 7$$

Whakaoti mō x, y

$$x=2$$

$$y=3$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$4x-y=5,-4x+5y=7$$

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

$$4x-y=5$$

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

$$4x=y+5$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x=\frac{1}{4}y+\frac{5}{4}$$

Whakakapia te \frac{5+y}{4} mō te x ki tērā atu whārite, -4x+5y=7.

$$-4\left(\frac{1}{4}y+\frac{5}{4}\right)+5y=7$$

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

$$-y-5+5y=7$$

Tāpiri -y ki te 5y.

$$4y-5=7$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$4y=12$$

Whakawehea ngā taha e rua ki te 4.

$$y=3$$

Whakaurua te 3 mō y ki x=\frac{1}{4}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\times 3+\frac{5}{4}$$

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

$$x=\frac{3+5}{4}$$

Tāpiri \frac{5}{4} ki te \frac{3}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$4x-y=5,-4x+5y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\-4&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\-4&5\end{matrix}\right))\left(\begin{matrix}5\\7\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-\left(-\left(-4\right)\right)}&-\frac{-1}{4\times 5-\left(-\left(-4\right)\right)}\\-\frac{-4}{4\times 5-\left(-\left(-4\right)\right)}&\frac{4}{4\times 5-\left(-\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}&\frac{1}{16}\\\frac{1}{4}&\frac{1}{4}\end{matrix}\right)\left(\begin{matrix}5\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{16}\times 5+\frac{1}{16}\times 7\\\frac{1}{4}\times 5+\frac{1}{4}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=2,y=3$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$4x-y=5,-4x+5y=7$$

Kia ōrite ai a 4x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-4\times 4x-4\left(-1\right)y=-4\times 5,4\left(-4\right)x+4\times 5y=4\times 7$$

Whakarūnātia.

$$-16x+4y=-20,-16x+20y=28$$

Me tango -16x+20y=28 mai i -16x+4y=-20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-16x+16x+4y-20y=-20-28$$

Tāpiri -16x ki te 16x. Ka whakakore atu ngā kupu -16x me 16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4y-20y=-20-28$$

Tāpiri 4y ki te -20y.

$$-16y=-20-28$$

Tāpiri -20 ki te -28.

$$-16y=-48$$

Whakawehea ngā taha e rua ki te -16.

$$y=3$$

Whakaurua te 3 mō y ki -4x+5y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+5\times 3=7$$

Whakareatia 5 ki te 3.

$$-4x+15=7$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-4x=-8$$

Whakawehea ngā taha e rua ki te -4.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=3$$