Whakaoti i te {l}{x/47+y=86}{x+y/25=49}

$$\left. \begin{array} { l } { \frac { x } { 47 } + y = 86 } \\ { x + \frac { y } { 25 } = 49 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{53533}{1174} = 45\frac{703}{1174} \approx 45.598807496$$

$$y = \frac{99825}{1174} = 85\frac{35}{1174} \approx 85.029812606$$

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.

$$\frac{1}{47}x+y=86,x+\frac{1}{25}y=49$$

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

$$\frac{1}{47}x+y=86$$

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

$$\frac{1}{47}x=-y+86$$

Me whakarea ngā taha e rua ki te 47.

$$x=47\left(-y+86\right)$$

Whakareatia 47 ki te -y+86.

$$x=-47y+4042$$

Whakakapia te -47y+4042 mō te x ki tērā atu whārite, x+\frac{1}{25}y=49.

$$-47y+4042+\frac{1}{25}y=49$$

Tāpiri -47y ki te \frac{y}{25}.

$$-\frac{1174}{25}y+4042=49$$

Me tango 4042 mai i ngā taha e rua o te whārite.

$$-\frac{1174}{25}y=-3993$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{1174}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{99825}{1174}$$

Whakaurua te \frac{99825}{1174} mō y ki x=-47y+4042. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-47\times \frac{99825}{1174}+4042$$

Whakareatia -47 ki te \frac{99825}{1174}.

$$x=-\frac{4691775}{1174}+4042$$

Tāpiri 4042 ki te -\frac{4691775}{1174}.

$$x=\frac{53533}{1174}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{53533}{1174},y=\frac{99825}{1174}$$

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.

$$\frac{1}{47}x+y=86,x+\frac{1}{25}y=49$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}86\\49\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right).

$$inverse(\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right))\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right))\left(\begin{matrix}86\\49\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right))\left(\begin{matrix}86\\49\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}\frac{1}{47}&1\\1&\frac{1}{25}\end{matrix}\right))\left(\begin{matrix}86\\49\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{25}}{\frac{1}{47}\times \frac{1}{25}-1}&-\frac{1}{\frac{1}{47}\times \frac{1}{25}-1}\\-\frac{1}{\frac{1}{47}\times \frac{1}{25}-1}&\frac{\frac{1}{47}}{\frac{1}{47}\times \frac{1}{25}-1}\end{matrix}\right)\left(\begin{matrix}86\\49\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{47}{1174}&\frac{1175}{1174}\\\frac{1175}{1174}&-\frac{25}{1174}\end{matrix}\right)\left(\begin{matrix}86\\49\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{47}{1174}\times 86+\frac{1175}{1174}\times 49\\\frac{1175}{1174}\times 86-\frac{25}{1174}\times 49\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{53533}{1174}\\\frac{99825}{1174}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{53533}{1174},y=\frac{99825}{1174}$$

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.

$$\frac{1}{47}x+y=86,x+\frac{1}{25}y=49$$

Kia ōrite ai a \frac{x}{47} 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 \frac{1}{47}.

$$\frac{1}{47}x+y=86,\frac{1}{47}x+\frac{1}{47}\times \frac{1}{25}y=\frac{1}{47}\times 49$$

Whakarūnātia.

$$\frac{1}{47}x+y=86,\frac{1}{47}x+\frac{1}{1175}y=\frac{49}{47}$$

Me tango \frac{1}{47}x+\frac{1}{1175}y=\frac{49}{47} mai i \frac{1}{47}x+y=86 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{1}{47}x-\frac{1}{47}x+y-\frac{1}{1175}y=86-\frac{49}{47}$$

Tāpiri \frac{x}{47} ki te -\frac{x}{47}. Ka whakakore atu ngā kupu \frac{x}{47} me -\frac{x}{47}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$y-\frac{1}{1175}y=86-\frac{49}{47}$$

Tāpiri y ki te -\frac{y}{1175}.

$$\frac{1174}{1175}y=86-\frac{49}{47}$$

Tāpiri 86 ki te -\frac{49}{47}.

$$\frac{1174}{1175}y=\frac{3993}{47}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{1174}{1175}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{99825}{1174}$$

Whakaurua te \frac{99825}{1174} mō y ki x+\frac{1}{25}y=49. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+\frac{1}{25}\times \frac{99825}{1174}=49$$

Whakareatia \frac{1}{25} ki te \frac{99825}{1174} 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.

$$x+\frac{3993}{1174}=49$$

Me tango \frac{3993}{1174} mai i ngā taha e rua o te whārite.

$$x=\frac{53533}{1174}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{53533}{1174},y=\frac{99825}{1174}$$