Whakaoti i te {l}{0.5x+0.7y=35}{x+0.4y=40}

$$\left\{ \begin{array} { l } { 0.5 x + 0.7 y = 35 } \\ { x + 0.4 y = 40 } \end{array} \right.$$

Whakaoti mō x, y

$$x=28$$

$$y=30$$

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.

$$0.5x+0.7y=35,x+0.4y=40$$

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

$$0.5x+0.7y=35$$

Me tango \frac{7y}{10} mai i ngā taha e rua o te whārite.

$$0.5x=-0.7y+35$$

Me whakarea ngā taha e rua ki te 2.

$$x=2\left(-0.7y+35\right)$$

Whakareatia 2 ki te -\frac{7y}{10}+35.

$$x=-1.4y+70$$

Whakakapia te -\frac{7y}{5}+70 mō te x ki tērā atu whārite, x+0.4y=40.

$$-1.4y+70+0.4y=40$$

Tāpiri -\frac{7y}{5} ki te \frac{2y}{5}.

$$-y+70=40$$

Me tango 70 mai i ngā taha e rua o te whārite.

$$-y=-30$$

Whakawehea ngā taha e rua ki te -1.

$$y=30$$

Whakaurua te 30 mō y ki x=-1.4y+70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-1.4\times 30+70$$

Whakareatia -1.4 ki te 30.

$$x=-42+70$$

Tāpiri 70 ki te -42.

$$x=28$$

Kua oti te pūnaha te whakatau.

$$x=28,y=30$$

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.

$$0.5x+0.7y=35,x+0.4y=40$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}35\\40\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right))\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.5&0.7\\1&0.4\end{matrix}\right))\left(\begin{matrix}35\\40\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.4}{0.5\times 0.4-0.7}&-\frac{0.7}{0.5\times 0.4-0.7}\\-\frac{1}{0.5\times 0.4-0.7}&\frac{0.5}{0.5\times 0.4-0.7}\end{matrix}\right)\left(\begin{matrix}35\\40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.8&1.4\\2&-1\end{matrix}\right)\left(\begin{matrix}35\\40\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-0.8\times 35+1.4\times 40\\2\times 35-40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\30\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=28,y=30$$

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.

$$0.5x+0.7y=35,x+0.4y=40$$

Kia ōrite ai a \frac{x}{2} me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 0.5.

$$0.5x+0.7y=35,0.5x+0.5\times 0.4y=0.5\times 40$$

Whakarūnātia.

$$0.5x+0.7y=35,0.5x+0.2y=20$$

Me tango 0.5x+0.2y=20 mai i 0.5x+0.7y=35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.5x-0.5x+0.7y-0.2y=35-20$$

Tāpiri \frac{x}{2} ki te -\frac{x}{2}. Ka whakakore atu ngā kupu \frac{x}{2} me -\frac{x}{2}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.7y-0.2y=35-20$$

Tāpiri \frac{7y}{10} ki te -\frac{y}{5}.

$$0.5y=35-20$$

Tāpiri 35 ki te -20.

$$0.5y=15$$

Me whakarea ngā taha e rua ki te 2.

$$y=30$$

Whakaurua te 30 mō y ki x+0.4y=40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+0.4\times 30=40$$

Whakareatia 0.4 ki te 30.

$$x+12=40$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$x=28$$

Kua oti te pūnaha te whakatau.

$$x=28,y=30$$