Whakaoti i te {l}{5x-8=y}{3x+2y=2}

$$\left. \begin{array} { l } { 5 x - 8 = y } \\ { 3 x + 2 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{18}{13} = 1\frac{5}{13} \approx 1.384615385$$

$$y = -\frac{14}{13} = -1\frac{1}{13} \approx -1.076923077$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$5x-8-y=0$$

Me tāpiri te 8 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x-y=8$$

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.

$$5x-y=8,3x+2y=2$$

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

$$5x-y=8$$

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

$$5x=y+8$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+8\right)$$

Whakareatia \frac{1}{5} ki te y+8.

$$x=\frac{1}{5}y+\frac{8}{5}$$

Whakakapia te \frac{8+y}{5} mō te x ki tērā atu whārite, 3x+2y=2.

$$3\left(\frac{1}{5}y+\frac{8}{5}\right)+2y=2$$

Whakareatia 3 ki te \frac{8+y}{5}.

$$\frac{3}{5}y+\frac{24}{5}+2y=2$$

Tāpiri \frac{3y}{5} ki te 2y.

$$\frac{13}{5}y+\frac{24}{5}=2$$

Me tango \frac{24}{5} mai i ngā taha e rua o te whārite.

$$\frac{13}{5}y=-\frac{14}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{13}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{14}{13}$$

Whakaurua te -\frac{14}{13} mō y ki x=\frac{1}{5}y+\frac{8}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\left(-\frac{14}{13}\right)+\frac{8}{5}$$

Whakareatia \frac{1}{5} ki te -\frac{14}{13} 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.

$$x=-\frac{14}{65}+\frac{8}{5}$$

Tāpiri \frac{8}{5} ki te -\frac{14}{65} 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.

$$x=\frac{18}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{13},y=-\frac{14}{13}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$5x-8-y=0$$

Me tāpiri te 8 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x-y=8$$

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

$$5x-y=8,3x+2y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\3&2\end{matrix}\right))\left(\begin{matrix}5&-1\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&2\end{matrix}\right))\left(\begin{matrix}8\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&2\end{matrix}\right))\left(\begin{matrix}8\\2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\3&2\end{matrix}\right))\left(\begin{matrix}8\\2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5\times 2-\left(-3\right)}&-\frac{-1}{5\times 2-\left(-3\right)}\\-\frac{3}{5\times 2-\left(-3\right)}&\frac{5}{5\times 2-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}8\\2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}&\frac{1}{13}\\-\frac{3}{13}&\frac{5}{13}\end{matrix}\right)\left(\begin{matrix}8\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{13}\times 8+\frac{1}{13}\times 2\\-\frac{3}{13}\times 8+\frac{5}{13}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{18}{13}\\-\frac{14}{13}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{18}{13},y=-\frac{14}{13}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$5x-8-y=0$$

Me tāpiri te 8 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x-y=8$$

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.

$$5x-y=8,3x+2y=2$$

Kia ōrite ai a 5x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$3\times 5x+3\left(-1\right)y=3\times 8,5\times 3x+5\times 2y=5\times 2$$

Whakarūnātia.

$$15x-3y=24,15x+10y=10$$

Me tango 15x+10y=10 mai i 15x-3y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x-3y-10y=24-10$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-10y=24-10$$

Tāpiri -3y ki te -10y.

$$-13y=24-10$$

Tāpiri 24 ki te -10.

$$-13y=14$$

Whakawehea ngā taha e rua ki te -13.

$$y=-\frac{14}{13}$$

Whakaurua te -\frac{14}{13} mō y ki 3x+2y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-\frac{14}{13}\right)=2$$

Whakareatia 2 ki te -\frac{14}{13}.

$$3x-\frac{28}{13}=2$$

Me tāpiri \frac{28}{13} ki ngā taha e rua o te whārite.

$$3x=\frac{54}{13}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{18}{13}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{18}{13},y=-\frac{14}{13}$$