Whakaoti i te {l}{8x-9y=15}{-5x+3y=9}

$$\left. \begin{array} { l } { 8 x - 9 y = 15 } \\ { - 5 x + 3 y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-6$$

$$y=-7$$

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.

$$8x-9y=15,-5x+3y=9$$

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

$$8x-9y=15$$

Me tāpiri 9y ki ngā taha e rua o te whārite.

$$8x=9y+15$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(9y+15\right)$$

Whakareatia \frac{1}{8} ki te 9y+15.

$$x=\frac{9}{8}y+\frac{15}{8}$$

Whakakapia te \frac{9y+15}{8} mō te x ki tērā atu whārite, -5x+3y=9.

$$-5\left(\frac{9}{8}y+\frac{15}{8}\right)+3y=9$$

Whakareatia -5 ki te \frac{9y+15}{8}.

$$-\frac{45}{8}y-\frac{75}{8}+3y=9$$

Tāpiri -\frac{45y}{8} ki te 3y.

$$-\frac{21}{8}y-\frac{75}{8}=9$$

Me tāpiri \frac{75}{8} ki ngā taha e rua o te whārite.

$$-\frac{21}{8}y=\frac{147}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{21}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-7$$

Whakaurua te -7 mō y ki x=\frac{9}{8}y+\frac{15}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{9}{8}\left(-7\right)+\frac{15}{8}$$

Whakareatia \frac{9}{8} ki te -7.

$$x=\frac{-63+15}{8}$$

Tāpiri \frac{15}{8} ki te -\frac{63}{8} 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.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=-7$$

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.

$$8x-9y=15,-5x+3y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-9\\-5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right))\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-9\\-5&3\end{matrix}\right))\left(\begin{matrix}15\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8\times 3-\left(-9\left(-5\right)\right)}&-\frac{-9}{8\times 3-\left(-9\left(-5\right)\right)}\\-\frac{-5}{8\times 3-\left(-9\left(-5\right)\right)}&\frac{8}{8\times 3-\left(-9\left(-5\right)\right)}\end{matrix}\right)\left(\begin{matrix}15\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&-\frac{3}{7}\\-\frac{5}{21}&-\frac{8}{21}\end{matrix}\right)\left(\begin{matrix}15\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 15-\frac{3}{7}\times 9\\-\frac{5}{21}\times 15-\frac{8}{21}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-6\\-7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-6,y=-7$$

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.

$$8x-9y=15,-5x+3y=9$$

Kia ōrite ai a 8x 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 8.

$$-5\times 8x-5\left(-9\right)y=-5\times 15,8\left(-5\right)x+8\times 3y=8\times 9$$

Whakarūnātia.

$$-40x+45y=-75,-40x+24y=72$$

Me tango -40x+24y=72 mai i -40x+45y=-75 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-40x+40x+45y-24y=-75-72$$

Tāpiri -40x ki te 40x. Ka whakakore atu ngā kupu -40x me 40x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$45y-24y=-75-72$$

Tāpiri 45y ki te -24y.

$$21y=-75-72$$

Tāpiri -75 ki te -72.

$$21y=-147$$

Whakawehea ngā taha e rua ki te 21.

$$y=-7$$

Whakaurua te -7 mō y ki -5x+3y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-5x+3\left(-7\right)=9$$

Whakareatia 3 ki te -7.

$$-5x-21=9$$

Me tāpiri 21 ki ngā taha e rua o te whārite.

$$-5x=30$$

Whakawehea ngā taha e rua ki te -5.

$$x=-6$$

Kua oti te pūnaha te whakatau.

$$x=-6,y=-7$$