Whakaoti i te {l}{4x-3y=5}{3x+2y=8}

$$\left. \begin{array} { l } { 4 x - 3 y = 5 } \\ { 3 x + 2 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=2$$

$$y=1$$

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.

$$4x-3y=5,3x+2y=8$$

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

$$4x-3y=5$$

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

$$4x=3y+5$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x=\frac{3}{4}y+\frac{5}{4}$$

Whakakapia te \frac{3y+5}{4} mō te x ki tērā atu whārite, 3x+2y=8.

$$3\left(\frac{3}{4}y+\frac{5}{4}\right)+2y=8$$

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

$$\frac{9}{4}y+\frac{15}{4}+2y=8$$

Tāpiri \frac{9y}{4} ki te 2y.

$$\frac{17}{4}y+\frac{15}{4}=8$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$\frac{17}{4}y=\frac{17}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{17}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{3}{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.

$$x=\frac{3+5}{4}$$

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.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$

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.

$$4x-3y=5,3x+2y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-3\\3&2\end{matrix}\right))\left(\begin{matrix}4&-3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{4\times 2-\left(-3\times 3\right)}&-\frac{-3}{4\times 2-\left(-3\times 3\right)}\\-\frac{3}{4\times 2-\left(-3\times 3\right)}&\frac{4}{4\times 2-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}5\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{3}{17}\\-\frac{3}{17}&\frac{4}{17}\end{matrix}\right)\left(\begin{matrix}5\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 5+\frac{3}{17}\times 8\\-\frac{3}{17}\times 5+\frac{4}{17}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=2,y=1$$

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.

$$4x-3y=5,3x+2y=8$$

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.

$$3\times 4x+3\left(-3\right)y=3\times 5,4\times 3x+4\times 2y=4\times 8$$

Whakarūnātia.

$$12x-9y=15,12x+8y=32$$

Me tango 12x+8y=32 mai i 12x-9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-9y-8y=15-32$$

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.

$$-9y-8y=15-32$$

Tāpiri -9y ki te -8y.

$$-17y=15-32$$

Tāpiri 15 ki te -32.

$$-17y=-17$$

Whakawehea ngā taha e rua ki te -17.

$$y=1$$

Whakaurua te 1 mō y ki 3x+2y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2=8$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$3x=6$$

Whakawehea ngā taha e rua ki te 3.

$$x=2$$

Kua oti te pūnaha te whakatau.

$$x=2,y=1$$