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

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

Whakaoti mō x, y

$$x = \frac{11}{7} = 1\frac{4}{7} \approx 1.571428571$$

$$y=\frac{6}{7}\approx 0.857142857$$

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.

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

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

$$x+4y=5$$

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

$$x=-4y+5$$

Whakakapia te -4y+5 mō te x ki tērā atu whārite, -2x-y=-4.

$$-2\left(-4y+5\right)-y=-4$$

Whakareatia -2 ki te -4y+5.

$$8y-10-y=-4$$

Tāpiri 8y ki te -y.

$$7y-10=-4$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$7y=6$$

Whakawehea ngā taha e rua ki te 7.

$$y=\frac{6}{7}$$

Whakaurua te \frac{6}{7} mō y ki x=-4y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-4\times \frac{6}{7}+5$$

Whakareatia -4 ki te \frac{6}{7}.

$$x=-\frac{24}{7}+5$$

Tāpiri 5 ki te -\frac{24}{7}.

$$x=\frac{11}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{7},y=\frac{6}{7}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&-\frac{4}{7}\\\frac{2}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}5\\-4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 5-\frac{4}{7}\left(-4\right)\\\frac{2}{7}\times 5+\frac{1}{7}\left(-4\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{7}\\\frac{6}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{11}{7},y=\frac{6}{7}$$

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.

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

Kia ōrite ai a x 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 1.

$$-2x-2\times 4y=-2\times 5,-2x-y=-4$$

Whakarūnātia.

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

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

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

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

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

Tāpiri -8y ki te y.

$$-7y=-10+4$$

Tāpiri -10 ki te 4.

$$-7y=-6$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{6}{7}$$

Whakaurua te \frac{6}{7} mō y ki -2x-y=-4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x-\frac{6}{7}=-4$$

Me tāpiri \frac{6}{7} ki ngā taha e rua o te whārite.

$$-2x=-\frac{22}{7}$$

Whakawehea ngā taha e rua ki te -2.

$$x=\frac{11}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{11}{7},y=\frac{6}{7}$$