Whakaoti i te {l}{8x-5y=10}{6x-4y=11}

$$\left. \begin{array} { l } { 8 x - 5 y = 10 } \\ { 6 x - 4 y = 11 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

$$y=-14$$

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.

$$8x-5y=10,6x-4y=11$$

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

$$8x-5y=10$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$8x=5y+10$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(5y+10\right)$$

Whakareatia \frac{1}{8} ki te 10+5y.

$$x=\frac{5}{8}y+\frac{5}{4}$$

Whakakapia te \frac{5}{4}+\frac{5y}{8} mō te x ki tērā atu whārite, 6x-4y=11.

$$6\left(\frac{5}{8}y+\frac{5}{4}\right)-4y=11$$

Whakareatia 6 ki te \frac{5}{4}+\frac{5y}{8}.

$$\frac{15}{4}y+\frac{15}{2}-4y=11$$

Tāpiri \frac{15y}{4} ki te -4y.

$$-\frac{1}{4}y+\frac{15}{2}=11$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-\frac{1}{4}y=\frac{7}{2}$$

Me whakarea ngā taha e rua ki te -4.

$$y=-14$$

Whakaurua te -14 mō y ki x=\frac{5}{8}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.

$$x=\frac{5}{8}\left(-14\right)+\frac{5}{4}$$

Whakareatia \frac{5}{8} ki te -14.

$$x=\frac{-35+5}{4}$$

Tāpiri \frac{5}{4} ki te -\frac{35}{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.

$$x=-\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{15}{2},y=-14$$

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.

$$8x-5y=10,6x-4y=11$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\11\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-5\\6&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&-5\\6&-4\end{matrix}\right))\left(\begin{matrix}10\\11\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{8\left(-4\right)-\left(-5\times 6\right)}&-\frac{-5}{8\left(-4\right)-\left(-5\times 6\right)}\\-\frac{6}{8\left(-4\right)-\left(-5\times 6\right)}&\frac{8}{8\left(-4\right)-\left(-5\times 6\right)}\end{matrix}\right)\left(\begin{matrix}10\\11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2&-\frac{5}{2}\\3&-4\end{matrix}\right)\left(\begin{matrix}10\\11\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\times 10-\frac{5}{2}\times 11\\3\times 10-4\times 11\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{2}\\-14\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{15}{2},y=-14$$

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.

$$8x-5y=10,6x-4y=11$$

Kia ōrite ai a 8x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

$$6\times 8x+6\left(-5\right)y=6\times 10,8\times 6x+8\left(-4\right)y=8\times 11$$

Whakarūnātia.

$$48x-30y=60,48x-32y=88$$

Me tango 48x-32y=88 mai i 48x-30y=60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$48x-48x-30y+32y=60-88$$

Tāpiri 48x ki te -48x. Ka whakakore atu ngā kupu 48x me -48x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-30y+32y=60-88$$

Tāpiri -30y ki te 32y.

$$2y=60-88$$

Tāpiri 60 ki te -88.

$$2y=-28$$

Whakawehea ngā taha e rua ki te 2.

$$y=-14$$

Whakaurua te -14 mō y ki 6x-4y=11. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x-4\left(-14\right)=11$$

Whakareatia -4 ki te -14.

$$6x+56=11$$

Me tango 56 mai i ngā taha e rua o te whārite.

$$6x=-45$$

Whakawehea ngā taha e rua ki te 6.

$$x=-\frac{15}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{15}{2},y=-14$$