Whakaoti i te {l}{4x+3y=71}{7x+5y=120}

$$\left\{ \begin{array} { l } { 4 x + 3 y = 71 } \\ { 7 x + 5 y = 120 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=17$$

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.

$$4x+3y=71,7x+5y=120$$

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

$$4x+3y=71$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$4x=-3y+71$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+71\right)$$

Whakareatia \frac{1}{4} ki te -3y+71.

$$x=-\frac{3}{4}y+\frac{71}{4}$$

Whakakapia te \frac{-3y+71}{4} mō te x ki tērā atu whārite, 7x+5y=120.

$$7\left(-\frac{3}{4}y+\frac{71}{4}\right)+5y=120$$

Whakareatia 7 ki te \frac{-3y+71}{4}.

$$-\frac{21}{4}y+\frac{497}{4}+5y=120$$

Tāpiri -\frac{21y}{4} ki te 5y.

$$-\frac{1}{4}y+\frac{497}{4}=120$$

Me tango \frac{497}{4} mai i ngā taha e rua o te whārite.

$$-\frac{1}{4}y=-\frac{17}{4}$$

Me whakarea ngā taha e rua ki te -4.

$$y=17$$

Whakaurua te 17 mō y ki x=-\frac{3}{4}y+\frac{71}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times 17+\frac{71}{4}$$

Whakareatia -\frac{3}{4} ki te 17.

$$x=\frac{-51+71}{4}$$

Tāpiri \frac{71}{4} ki te -\frac{51}{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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=17$$

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.

$$4x+3y=71,7x+5y=120$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}71\\120\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\7&5\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&3\\7&5\end{matrix}\right))\left(\begin{matrix}4&3\\7&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\7&5\end{matrix}\right))\left(\begin{matrix}71\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\7&5\end{matrix}\right))\left(\begin{matrix}71\\120\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\7&5\end{matrix}\right))\left(\begin{matrix}71\\120\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-3\times 7}&-\frac{3}{4\times 5-3\times 7}\\-\frac{7}{4\times 5-3\times 7}&\frac{4}{4\times 5-3\times 7}\end{matrix}\right)\left(\begin{matrix}71\\120\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5&3\\7&-4\end{matrix}\right)\left(\begin{matrix}71\\120\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\times 71+3\times 120\\7\times 71-4\times 120\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\17\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=17$$

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.

$$4x+3y=71,7x+5y=120$$

Kia ōrite ai a 4x 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 4.

$$7\times 4x+7\times 3y=7\times 71,4\times 7x+4\times 5y=4\times 120$$

Whakarūnātia.

$$28x+21y=497,28x+20y=480$$

Me tango 28x+20y=480 mai i 28x+21y=497 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x+21y-20y=497-480$$

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.

$$21y-20y=497-480$$

Tāpiri 21y ki te -20y.

$$y=497-480$$

Tāpiri 497 ki te -480.

$$y=17$$

Whakaurua te 17 mō y ki 7x+5y=120. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x+5\times 17=120$$

Whakareatia 5 ki te 17.

$$7x+85=120$$

Me tango 85 mai i ngā taha e rua o te whārite.

$$7x=35$$

Whakawehea ngā taha e rua ki te 7.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=17$$