Whakaoti i te {l}{5y+3=9x}{4x-2=1y}

$$\left. \begin{array} { l } { 5 y + 3 = 9 x } \\ { 4 x - 2 = 1 y } \end{array} \right.$$

Whakaoti mō y, x

$$x=\frac{7}{11}\approx 0.636363636$$

$$y=\frac{6}{11}\approx 0.545454545$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$5y+3-9x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5y-9x=-3$$

Whakaarohia te whārite tuarua. Tangohia te 1y mai i ngā taha e rua.

$$4x-2-y=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4x-y=2$$

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.

$$5y-9x=-3,-y+4x=2$$

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

$$5y-9x=-3$$

Me tāpiri 9x ki ngā taha e rua o te whārite.

$$5y=9x-3$$

Whakawehea ngā taha e rua ki te 5.

$$y=\frac{1}{5}\left(9x-3\right)$$

Whakareatia \frac{1}{5} ki te 9x-3.

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

Whakakapia te \frac{9x-3}{5} mō te y ki tērā atu whārite, -y+4x=2.

$$-\left(\frac{9}{5}x-\frac{3}{5}\right)+4x=2$$

Whakareatia -1 ki te \frac{9x-3}{5}.

$$-\frac{9}{5}x+\frac{3}{5}+4x=2$$

Tāpiri -\frac{9x}{5} ki te 4x.

$$\frac{11}{5}x+\frac{3}{5}=2$$

Me tango \frac{3}{5} mai i ngā taha e rua o te whārite.

$$\frac{11}{5}x=\frac{7}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{7}{11}$$

Whakaurua te \frac{7}{11} mō x ki y=\frac{9}{5}x-\frac{3}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{9}{5}\times \frac{7}{11}-\frac{3}{5}$$

Whakareatia \frac{9}{5} ki te \frac{7}{11} 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.

$$y=\frac{63}{55}-\frac{3}{5}$$

Tāpiri -\frac{3}{5} ki te \frac{63}{55} 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.

$$y=\frac{6}{11}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{6}{11},x=\frac{7}{11}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$5y+3-9x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5y-9x=-3$$

Whakaarohia te whārite tuarua. Tangohia te 1y mai i ngā taha e rua.

$$4x-2-y=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4x-y=2$$

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

$$5y-9x=-3,-y+4x=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-9\\-1&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right))\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right))\left(\begin{matrix}-3\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right))\left(\begin{matrix}-3\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}5&-9\\-1&4\end{matrix}\right))\left(\begin{matrix}-3\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{5\times 4-\left(-9\left(-1\right)\right)}&-\frac{-9}{5\times 4-\left(-9\left(-1\right)\right)}\\-\frac{-1}{5\times 4-\left(-9\left(-1\right)\right)}&\frac{5}{5\times 4-\left(-9\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}&\frac{9}{11}\\\frac{1}{11}&\frac{5}{11}\end{matrix}\right)\left(\begin{matrix}-3\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{11}\left(-3\right)+\frac{9}{11}\times 2\\\frac{1}{11}\left(-3\right)+\frac{5}{11}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{6}{11}\\\frac{7}{11}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=\frac{6}{11},x=\frac{7}{11}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 9x mai i ngā taha e rua.

$$5y+3-9x=0$$

Tangohia te 3 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5y-9x=-3$$

Whakaarohia te whārite tuarua. Tangohia te 1y mai i ngā taha e rua.

$$4x-2-y=0$$

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4x-y=2$$

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.

$$5y-9x=-3,-y+4x=2$$

Kia ōrite ai a 5y me -y, 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 5.

$$-5y-\left(-9x\right)=-\left(-3\right),5\left(-1\right)y+5\times 4x=5\times 2$$

Whakarūnātia.

$$-5y+9x=3,-5y+20x=10$$

Me tango -5y+20x=10 mai i -5y+9x=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5y+5y+9x-20x=3-10$$

Tāpiri -5y ki te 5y. Ka whakakore atu ngā kupu -5y me 5y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9x-20x=3-10$$

Tāpiri 9x ki te -20x.

$$-11x=3-10$$

Tāpiri 3 ki te -10.

$$-11x=-7$$

Whakawehea ngā taha e rua ki te -11.

$$x=\frac{7}{11}$$

Whakaurua te \frac{7}{11} mō x ki -y+4x=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-y+4\times \frac{7}{11}=2$$

Whakareatia 4 ki te \frac{7}{11}.

$$-y+\frac{28}{11}=2$$

Me tango \frac{28}{11} mai i ngā taha e rua o te whārite.

$$-y=-\frac{6}{11}$$

Whakawehea ngā taha e rua ki te -1.

$$y=\frac{6}{11}$$

Kua oti te pūnaha te whakatau.

$$y=\frac{6}{11},x=\frac{7}{11}$$