Whakaoti i te {l}{4x-3y=1}{5x+y=7}

$$\left. \begin{array} { l } { 4 x - 3 y = 1 } \\ { 5 x + y = 7 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{22}{19} = 1\frac{3}{19} \approx 1.157894737$$

$$y = \frac{23}{19} = 1\frac{4}{19} \approx 1.210526316$$

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.

$$4x-3y=1,5x+y=7$$

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

$$4x-3y=1$$

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

$$4x=3y+1$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(3y+1\right)$$

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

$$x=\frac{3}{4}y+\frac{1}{4}$$

Whakakapia te \frac{3y+1}{4} mō te x ki tērā atu whārite, 5x+y=7.

$$5\left(\frac{3}{4}y+\frac{1}{4}\right)+y=7$$

Whakareatia 5 ki te \frac{3y+1}{4}.

$$\frac{15}{4}y+\frac{5}{4}+y=7$$

Tāpiri \frac{15y}{4} ki te y.

$$\frac{19}{4}y+\frac{5}{4}=7$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$\frac{19}{4}y=\frac{23}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{23}{19}$$

Whakaurua te \frac{23}{19} mō y ki x=\frac{3}{4}y+\frac{1}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{4}\times \frac{23}{19}+\frac{1}{4}$$

Whakareatia \frac{3}{4} ki te \frac{23}{19} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{69}{76}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te \frac{69}{76} 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.

$$x=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{19},y=\frac{23}{19}$$

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.

$$4x-3y=1,5x+y=7$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-3\\5&1\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\5&1\end{matrix}\right))\left(\begin{matrix}1\\7\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}&\frac{3}{19}\\-\frac{5}{19}&\frac{4}{19}\end{matrix}\right)\left(\begin{matrix}1\\7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{19}+\frac{3}{19}\times 7\\-\frac{5}{19}+\frac{4}{19}\times 7\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{19}\\\frac{23}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{22}{19},y=\frac{23}{19}$$

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.

$$4x-3y=1,5x+y=7$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\left(-3\right)y=5,4\times 5x+4y=4\times 7$$

Whakarūnātia.

$$20x-15y=5,20x+4y=28$$

Me tango 20x+4y=28 mai i 20x-15y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-15y-4y=5-28$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-15y-4y=5-28$$

Tāpiri -15y ki te -4y.

$$-19y=5-28$$

Tāpiri 5 ki te -28.

$$-19y=-23$$

Whakawehea ngā taha e rua ki te -19.

$$y=\frac{23}{19}$$

Whakaurua te \frac{23}{19} mō y ki 5x+y=7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+\frac{23}{19}=7$$

Me tango \frac{23}{19} mai i ngā taha e rua o te whārite.

$$5x=\frac{110}{19}$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{22}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{19},y=\frac{23}{19}$$