Whakaoti i te {l}{4x+y=7}{3x+2y=9}

$$\left. \begin{array} { l } { 4 x + y = 7 } \\ { 3 x + 2 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=3$$

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.

$$4x+y=7,3x+2y=9$$

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

$$4x+y=7$$

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

$$4x=-y+7$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te -y+7.

$$x=-\frac{1}{4}y+\frac{7}{4}$$

Whakakapia te \frac{-y+7}{4} mō te x ki tērā atu whārite, 3x+2y=9.

$$3\left(-\frac{1}{4}y+\frac{7}{4}\right)+2y=9$$

Whakareatia 3 ki te \frac{-y+7}{4}.

$$-\frac{3}{4}y+\frac{21}{4}+2y=9$$

Tāpiri -\frac{3y}{4} ki te 2y.

$$\frac{5}{4}y+\frac{21}{4}=9$$

Me tango \frac{21}{4} mai i ngā taha e rua o te whārite.

$$\frac{5}{4}y=\frac{15}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, 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{7}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\times 3+\frac{7}{4}$$

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

$$x=\frac{-3+7}{4}$$

Tāpiri \frac{7}{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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$

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.

$$4x+y=7,3x+2y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&1\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&1\\3&2\end{matrix}\right))\left(\begin{matrix}4&1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\3&2\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\3&2\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&1\\3&2\end{matrix}\right))\left(\begin{matrix}7\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-3}&-\frac{1}{4\times 2-3}\\-\frac{3}{4\times 2-3}&\frac{4}{4\times 2-3}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\-\frac{3}{5}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}7\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 7-\frac{1}{5}\times 9\\-\frac{3}{5}\times 7+\frac{4}{5}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=3$$

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.

$$4x+y=7,3x+2y=9$$

Kia ōrite ai a 4x me 3x, 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 4.

$$3\times 4x+3y=3\times 7,4\times 3x+4\times 2y=4\times 9$$

Whakarūnātia.

$$12x+3y=21,12x+8y=36$$

Me tango 12x+8y=36 mai i 12x+3y=21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+3y-8y=21-36$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-8y=21-36$$

Tāpiri 3y ki te -8y.

$$-5y=21-36$$

Tāpiri 21 ki te -36.

$$-5y=-15$$

Whakawehea ngā taha e rua ki te -5.

$$y=3$$

Whakaurua te 3 mō y ki 3x+2y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times 3=9$$

Whakareatia 2 ki te 3.

$$3x+6=9$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x=3$$

Whakawehea ngā taha e rua ki te 3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=3$$