Whakaoti i te {l}{4x+y=5}{2x-y=-2}

$$\left\{ \begin{array} { l } { 4 x + y = 5 } \\ { 2 x - y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{2}=0.5$$

$$y=3$$

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.

$$4x+y=5,2x-y=-2$$

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 tango y mai i 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{-y+5}{4} mō te x ki tērā atu whārite, 2x-y=-2.

$$2\left(-\frac{1}{4}y+\frac{5}{4}\right)-y=-2$$

Whakareatia 2 ki te \frac{-y+5}{4}.

$$-\frac{1}{2}y+\frac{5}{2}-y=-2$$

Tāpiri -\frac{y}{2} ki te -y.

$$-\frac{3}{2}y+\frac{5}{2}=-2$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$-\frac{3}{2}y=-\frac{9}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, 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{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.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=3$$

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.

$$4x+y=5,2x-y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\2&-1\end{matrix}\right))\left(\begin{matrix}4&1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&-1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&-1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\2&-1\end{matrix}\right))\left(\begin{matrix}5\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-2}&-\frac{1}{4\left(-1\right)-2}\\-\frac{2}{4\left(-1\right)-2}&\frac{4}{4\left(-1\right)-2}\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{1}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 5+\frac{1}{6}\left(-2\right)\\\frac{1}{3}\times 5-\frac{2}{3}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{2},y=3$$

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.

$$4x+y=5,2x-y=-2$$

Kia ōrite ai a 4x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$2\times 4x+2y=2\times 5,4\times 2x+4\left(-1\right)y=4\left(-2\right)$$

Whakarūnātia.

$$8x+2y=10,8x-4y=-8$$

Me tango 8x-4y=-8 mai i 8x+2y=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+2y+4y=10+8$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$2y+4y=10+8$$

Tāpiri 2y ki te 4y.

$$6y=10+8$$

Tāpiri 10 ki te 8.

$$6y=18$$

Whakawehea ngā taha e rua ki te 6.

$$y=3$$

Whakaurua te 3 mō y ki 2x-y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3=-2$$

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

$$2x=1$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{2},y=3$$