Whakaoti i te {l}{-8x+4y=24}{-7x+7y=28}

$$\left\{ \begin{array} { l } { - 8 x + 4 y = 24 } \\ { - 7 x + 7 y = 28 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-2$$

$$y=2$$

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.

$$-8x+4y=24,-7x+7y=28$$

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

$$-8x+4y=24$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$-8x=-4y+24$$

Whakawehea ngā taha e rua ki te -8.

$$x=-\frac{1}{8}\left(-4y+24\right)$$

Whakareatia -\frac{1}{8} ki te -4y+24.

$$x=\frac{1}{2}y-3$$

Whakakapia te \frac{y}{2}-3 mō te x ki tērā atu whārite, -7x+7y=28.

$$-7\left(\frac{1}{2}y-3\right)+7y=28$$

Whakareatia -7 ki te \frac{y}{2}-3.

$$-\frac{7}{2}y+21+7y=28$$

Tāpiri -\frac{7y}{2} ki te 7y.

$$\frac{7}{2}y+21=28$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$\frac{7}{2}y=7$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

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

$$x=\frac{1}{2}\times 2-3$$

Whakareatia \frac{1}{2} ki te 2.

$$x=1-3$$

Tāpiri -3 ki te 1.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$

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.

$$-8x+4y=24,-7x+7y=28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\28\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-8&4\\-7&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right))\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right))\left(\begin{matrix}24\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right))\left(\begin{matrix}24\\28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-8&4\\-7&7\end{matrix}\right))\left(\begin{matrix}24\\28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{-8\times 7-4\left(-7\right)}&-\frac{4}{-8\times 7-4\left(-7\right)}\\-\frac{-7}{-8\times 7-4\left(-7\right)}&-\frac{8}{-8\times 7-4\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}24\\28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{7}\\-\frac{1}{4}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}24\\28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 24+\frac{1}{7}\times 28\\-\frac{1}{4}\times 24+\frac{2}{7}\times 28\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-2,y=2$$

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.

$$-8x+4y=24,-7x+7y=28$$

Kia ōrite ai a -8x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -8.

$$-7\left(-8\right)x-7\times 4y=-7\times 24,-8\left(-7\right)x-8\times 7y=-8\times 28$$

Whakarūnātia.

$$56x-28y=-168,56x-56y=-224$$

Me tango 56x-56y=-224 mai i 56x-28y=-168 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$56x-56x-28y+56y=-168+224$$

Tāpiri 56x ki te -56x. Ka whakakore atu ngā kupu 56x me -56x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-28y+56y=-168+224$$

Tāpiri -28y ki te 56y.

$$28y=-168+224$$

Tāpiri -168 ki te 224.

$$28y=56$$

Whakawehea ngā taha e rua ki te 28.

$$y=2$$

Whakaurua te 2 mō y ki -7x+7y=28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+7\times 2=28$$

Whakareatia 7 ki te 2.

$$-7x+14=28$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$-7x=14$$

Whakawehea ngā taha e rua ki te -7.

$$x=-2$$

Kua oti te pūnaha te whakatau.

$$x=-2,y=2$$