Whakaoti i te {l}{4x=9-3y}{5y=12-5x}

$$\left. \begin{array} { l } { 4 x = 9 - 3 y } \\ { 5 y = 12 - 5 x } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

$$y=\frac{3}{5}=0.6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 3y ki ngā taha e rua.

$$4x+3y=9$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$5y+5x=12$$

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.

$$4x+3y=9,5x+5y=12$$

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

$$4x+3y=9$$

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

$$4x=-3y+9$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+9\right)$$

Whakareatia \frac{1}{4} ki te -3y+9.

$$x=-\frac{3}{4}y+\frac{9}{4}$$

Whakakapia te \frac{-3y+9}{4} mō te x ki tērā atu whārite, 5x+5y=12.

$$5\left(-\frac{3}{4}y+\frac{9}{4}\right)+5y=12$$

Whakareatia 5 ki te \frac{-3y+9}{4}.

$$-\frac{15}{4}y+\frac{45}{4}+5y=12$$

Tāpiri -\frac{15y}{4} ki te 5y.

$$\frac{5}{4}y+\frac{45}{4}=12$$

Me tango \frac{45}{4} mai i ngā taha e rua o te whārite.

$$\frac{5}{4}y=\frac{3}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{3}{5}$$

Whakaurua te \frac{3}{5} mō y ki x=-\frac{3}{4}y+\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times \frac{3}{5}+\frac{9}{4}$$

Whakareatia -\frac{3}{4} ki te \frac{3}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{9}{20}+\frac{9}{4}$$

Tāpiri \frac{9}{4} ki te -\frac{9}{20} 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.

$$x=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{5},y=\frac{3}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 3y ki ngā taha e rua.

$$4x+3y=9$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$5y+5x=12$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$4x+3y=9,5x+5y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\12\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&3\\5&5\end{matrix}\right))\left(\begin{matrix}4&3\\5&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\5&5\end{matrix}\right))\left(\begin{matrix}9\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{4\times 5-3\times 5}&-\frac{3}{4\times 5-3\times 5}\\-\frac{5}{4\times 5-3\times 5}&\frac{4}{4\times 5-3\times 5}\end{matrix}\right)\left(\begin{matrix}9\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-\frac{3}{5}\\-1&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}9\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9-\frac{3}{5}\times 12\\-9+\frac{4}{5}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{5}\\\frac{3}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{9}{5},y=\frac{3}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 3y ki ngā taha e rua.

$$4x+3y=9$$

Whakaarohia te whārite tuarua. Me tāpiri te 5x ki ngā taha e rua.

$$5y+5x=12$$

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.

$$4x+3y=9,5x+5y=12$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\times 3y=5\times 9,4\times 5x+4\times 5y=4\times 12$$

Whakarūnātia.

$$20x+15y=45,20x+20y=48$$

Me tango 20x+20y=48 mai i 20x+15y=45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+15y-20y=45-48$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-20y=45-48$$

Tāpiri 15y ki te -20y.

$$-5y=45-48$$

Tāpiri 45 ki te -48.

$$-5y=-3$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{3}{5}$$

Whakaurua te \frac{3}{5} mō y ki 5x+5y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+5\times \frac{3}{5}=12$$

Whakareatia 5 ki te \frac{3}{5}.

$$5x+3=12$$

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

$$5x=9$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{9}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{9}{5},y=\frac{3}{5}$$