Whakaoti i te {l}{3x-8y=9}{4x+3y=-10}

$$\left\{ \begin{array} { l } { 3 x - 8 y = 9 } \\ { 4 x + 3 y = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{53}{41} = -1\frac{12}{41} \approx -1.292682927$$

$$y = -\frac{66}{41} = -1\frac{25}{41} \approx -1.609756098$$

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.

$$3x-8y=9,4x+3y=-10$$

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

$$3x-8y=9$$

Me tāpiri 8y ki ngā taha e rua o te whārite.

$$3x=8y+9$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(8y+9\right)$$

Whakareatia \frac{1}{3} ki te 8y+9.

$$x=\frac{8}{3}y+3$$

Whakakapia te \frac{8y}{3}+3 mō te x ki tērā atu whārite, 4x+3y=-10.

$$4\left(\frac{8}{3}y+3\right)+3y=-10$$

Whakareatia 4 ki te \frac{8y}{3}+3.

$$\frac{32}{3}y+12+3y=-10$$

Tāpiri \frac{32y}{3} ki te 3y.

$$\frac{41}{3}y+12=-10$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$\frac{41}{3}y=-22$$

Whakawehea ngā taha e rua o te whārite ki te \frac{41}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{66}{41}$$

Whakaurua te -\frac{66}{41} mō y ki x=\frac{8}{3}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{3}\left(-\frac{66}{41}\right)+3$$

Whakareatia \frac{8}{3} ki te -\frac{66}{41} 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.

$$x=-\frac{176}{41}+3$$

Tāpiri 3 ki te -\frac{176}{41}.

$$x=-\frac{53}{41}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{53}{41},y=-\frac{66}{41}$$

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.

$$3x-8y=9,4x+3y=-10$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-8\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-8\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-8\\4&3\end{matrix}\right))\left(\begin{matrix}3&-8\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-8\\4&3\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-8\\4&3\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-8\\4&3\end{matrix}\right))\left(\begin{matrix}9\\-10\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-\left(-8\times 4\right)}&-\frac{-8}{3\times 3-\left(-8\times 4\right)}\\-\frac{4}{3\times 3-\left(-8\times 4\right)}&\frac{3}{3\times 3-\left(-8\times 4\right)}\end{matrix}\right)\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{41}&\frac{8}{41}\\-\frac{4}{41}&\frac{3}{41}\end{matrix}\right)\left(\begin{matrix}9\\-10\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{41}\times 9+\frac{8}{41}\left(-10\right)\\-\frac{4}{41}\times 9+\frac{3}{41}\left(-10\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{53}{41}\\-\frac{66}{41}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{53}{41},y=-\frac{66}{41}$$

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.

$$3x-8y=9,4x+3y=-10$$

Kia ōrite ai a 3x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$4\times 3x+4\left(-8\right)y=4\times 9,3\times 4x+3\times 3y=3\left(-10\right)$$

Whakarūnātia.

$$12x-32y=36,12x+9y=-30$$

Me tango 12x+9y=-30 mai i 12x-32y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-32y-9y=36+30$$

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.

$$-32y-9y=36+30$$

Tāpiri -32y ki te -9y.

$$-41y=36+30$$

Tāpiri 36 ki te 30.

$$-41y=66$$

Whakawehea ngā taha e rua ki te -41.

$$y=-\frac{66}{41}$$

Whakaurua te -\frac{66}{41} mō y ki 4x+3y=-10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+3\left(-\frac{66}{41}\right)=-10$$

Whakareatia 3 ki te -\frac{66}{41}.

$$4x-\frac{198}{41}=-10$$

Me tāpiri \frac{198}{41} ki ngā taha e rua o te whārite.

$$4x=-\frac{212}{41}$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{53}{41}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{53}{41},y=-\frac{66}{41}$$