Whakaoti i te {l}{7x+4y=52}{4x-4y=-8}

$$\left. \begin{array} { l } { 7 x + 4 y = 52 } \\ { 4 x - 4 y = - 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=6$$

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.

$$7x+4y=52,4x-4y=-8$$

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

$$7x+4y=52$$

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

$$7x=-4y+52$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-4y+52\right)$$

Whakareatia \frac{1}{7} ki te -4y+52.

$$x=-\frac{4}{7}y+\frac{52}{7}$$

Whakakapia te \frac{-4y+52}{7} mō te x ki tērā atu whārite, 4x-4y=-8.

$$4\left(-\frac{4}{7}y+\frac{52}{7}\right)-4y=-8$$

Whakareatia 4 ki te \frac{-4y+52}{7}.

$$-\frac{16}{7}y+\frac{208}{7}-4y=-8$$

Tāpiri -\frac{16y}{7} ki te -4y.

$$-\frac{44}{7}y+\frac{208}{7}=-8$$

Me tango \frac{208}{7} mai i ngā taha e rua o te whārite.

$$-\frac{44}{7}y=-\frac{264}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{44}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=6$$

Whakaurua te 6 mō y ki x=-\frac{4}{7}y+\frac{52}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{7}\times 6+\frac{52}{7}$$

Whakareatia -\frac{4}{7} ki te 6.

$$x=\frac{-24+52}{7}$$

Tāpiri \frac{52}{7} ki te -\frac{24}{7} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=6$$

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.

$$7x+4y=52,4x-4y=-8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&4\\4&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}52\\-8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}7&4\\4&-4\end{matrix}\right))\left(\begin{matrix}7&4\\4&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&4\\4&-4\end{matrix}\right))\left(\begin{matrix}52\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&4\\4&-4\end{matrix}\right))\left(\begin{matrix}52\\-8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&4\\4&-4\end{matrix}\right))\left(\begin{matrix}52\\-8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{7\left(-4\right)-4\times 4}&-\frac{4}{7\left(-4\right)-4\times 4}\\-\frac{4}{7\left(-4\right)-4\times 4}&\frac{7}{7\left(-4\right)-4\times 4}\end{matrix}\right)\left(\begin{matrix}52\\-8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}&\frac{1}{11}\\\frac{1}{11}&-\frac{7}{44}\end{matrix}\right)\left(\begin{matrix}52\\-8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{11}\times 52+\frac{1}{11}\left(-8\right)\\\frac{1}{11}\times 52-\frac{7}{44}\left(-8\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=6$$

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.

$$7x+4y=52,4x-4y=-8$$

Kia ōrite ai a 7x 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 7.

$$4\times 7x+4\times 4y=4\times 52,7\times 4x+7\left(-4\right)y=7\left(-8\right)$$

Whakarūnātia.

$$28x+16y=208,28x-28y=-56$$

Me tango 28x-28y=-56 mai i 28x+16y=208 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x+16y+28y=208+56$$

Tāpiri 28x ki te -28x. Ka whakakore atu ngā kupu 28x me -28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$16y+28y=208+56$$

Tāpiri 16y ki te 28y.

$$44y=208+56$$

Tāpiri 208 ki te 56.

$$44y=264$$

Whakawehea ngā taha e rua ki te 44.

$$y=6$$

Whakaurua te 6 mō y ki 4x-4y=-8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-4\times 6=-8$$

Whakareatia -4 ki te 6.

$$4x-24=-8$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$4x=16$$

Whakawehea ngā taha e rua ki te 4.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=6$$