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

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

Whakaoti mō x, y

$$x = \frac{25}{17} = 1\frac{8}{17} \approx 1.470588235$$

$$y=\frac{12}{17}\approx 0.705882353$$

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.

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

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

$$x+5y=5$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$x=-5y+5$$

Whakakapia te -5y+5 mō te x ki tērā atu whārite, 3x-2y=3.

$$3\left(-5y+5\right)-2y=3$$

Whakareatia 3 ki te -5y+5.

$$-15y+15-2y=3$$

Tāpiri -15y ki te -2y.

$$-17y+15=3$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-17y=-12$$

Whakawehea ngā taha e rua ki te -17.

$$y=\frac{12}{17}$$

Whakaurua te \frac{12}{17} mō y ki x=-5y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-5\times \frac{12}{17}+5$$

Whakareatia -5 ki te \frac{12}{17}.

$$x=-\frac{60}{17}+5$$

Tāpiri 5 ki te -\frac{60}{17}.

$$x=\frac{25}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{17},y=\frac{12}{17}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}&\frac{5}{17}\\\frac{3}{17}&-\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}5\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{17}\times 5+\frac{5}{17}\times 3\\\frac{3}{17}\times 5-\frac{1}{17}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{25}{17}\\\frac{12}{17}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{25}{17},y=\frac{12}{17}$$

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.

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

Kia ōrite ai a x 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 1.

$$3x+3\times 5y=3\times 5,3x-2y=3$$

Whakarūnātia.

$$3x+15y=15,3x-2y=3$$

Me tango 3x-2y=3 mai i 3x+15y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+15y+2y=15-3$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y+2y=15-3$$

Tāpiri 15y ki te 2y.

$$17y=15-3$$

Tāpiri 15 ki te -3.

$$17y=12$$

Whakawehea ngā taha e rua ki te 17.

$$y=\frac{12}{17}$$

Whakaurua te \frac{12}{17} mō y ki 3x-2y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-2\times \frac{12}{17}=3$$

Whakareatia -2 ki te \frac{12}{17}.

$$3x-\frac{24}{17}=3$$

Me tāpiri \frac{24}{17} ki ngā taha e rua o te whārite.

$$3x=\frac{75}{17}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{25}{17}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{25}{17},y=\frac{12}{17}$$