Whakaoti i te {l}{3x+5y=21}{5x+2y=4}

$$\left. \begin{array} { l } { 3 x + 5 y = 21 } \\ { 5 x + 2 y = 4 } \end{array} \right.$$

Whakaoti mō x, y

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

$$y = \frac{93}{19} = 4\frac{17}{19} \approx 4.894736842$$

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.

$$3x+5y=21,5x+2y=4$$

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

$$3x+5y=21$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$3x=-5y+21$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y+21\right)$$

Whakareatia \frac{1}{3} ki te -5y+21.

$$x=-\frac{5}{3}y+7$$

Whakakapia te -\frac{5y}{3}+7 mō te x ki tērā atu whārite, 5x+2y=4.

$$5\left(-\frac{5}{3}y+7\right)+2y=4$$

Whakareatia 5 ki te -\frac{5y}{3}+7.

$$-\frac{25}{3}y+35+2y=4$$

Tāpiri -\frac{25y}{3} ki te 2y.

$$-\frac{19}{3}y+35=4$$

Me tango 35 mai i ngā taha e rua o te whārite.

$$-\frac{19}{3}y=-31$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{93}{19}$$

Whakaurua te \frac{93}{19} mō y ki x=-\frac{5}{3}y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\times \frac{93}{19}+7$$

Whakareatia -\frac{5}{3} ki te \frac{93}{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.

$$x=-\frac{155}{19}+7$$

Tāpiri 7 ki te -\frac{155}{19}.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{22}{19},y=\frac{93}{19}$$

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.

$$3x+5y=21,5x+2y=4$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}21\\4\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\5&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\5&2\end{matrix}\right))\left(\begin{matrix}3&5\\5&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&2\end{matrix}\right))\left(\begin{matrix}21\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&2\end{matrix}\right))\left(\begin{matrix}21\\4\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\5&2\end{matrix}\right))\left(\begin{matrix}21\\4\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{3\times 2-5\times 5}&-\frac{5}{3\times 2-5\times 5}\\-\frac{5}{3\times 2-5\times 5}&\frac{3}{3\times 2-5\times 5}\end{matrix}\right)\left(\begin{matrix}21\\4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{19}&\frac{5}{19}\\\frac{5}{19}&-\frac{3}{19}\end{matrix}\right)\left(\begin{matrix}21\\4\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{19}\times 21+\frac{5}{19}\times 4\\\frac{5}{19}\times 21-\frac{3}{19}\times 4\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{22}{19}\\\frac{93}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{22}{19},y=\frac{93}{19}$$

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.

$$3x+5y=21,5x+2y=4$$

Kia ōrite ai a 3x 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 3.

$$5\times 3x+5\times 5y=5\times 21,3\times 5x+3\times 2y=3\times 4$$

Whakarūnātia.

$$15x+25y=105,15x+6y=12$$

Me tango 15x+6y=12 mai i 15x+25y=105 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+25y-6y=105-12$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y-6y=105-12$$

Tāpiri 25y ki te -6y.

$$19y=105-12$$

Tāpiri 105 ki te -12.

$$19y=93$$

Whakawehea ngā taha e rua ki te 19.

$$y=\frac{93}{19}$$

Whakaurua te \frac{93}{19} mō y ki 5x+2y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+2\times \frac{93}{19}=4$$

Whakareatia 2 ki te \frac{93}{19}.

$$5x+\frac{186}{19}=4$$

Me tango \frac{186}{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.

$$x=-\frac{22}{19},y=\frac{93}{19}$$