Whakaoti i te {l}{70x+190y=2035}{x+y=14.5}

$$\left. \begin{array}{l}{ 70 x + 190 y = 2035 }\\{ x + y = 14.5 }\end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=8.5$$

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.

$$70x+190y=2035,x+y=14.5$$

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

$$70x+190y=2035$$

Me tango 190y mai i ngā taha e rua o te whārite.

$$70x=-190y+2035$$

Whakawehea ngā taha e rua ki te 70.

$$x=\frac{1}{70}\left(-190y+2035\right)$$

Whakareatia \frac{1}{70} ki te -190y+2035.

$$x=-\frac{19}{7}y+\frac{407}{14}$$

Whakakapia te -\frac{19y}{7}+\frac{407}{14} mō te x ki tērā atu whārite, x+y=14.5.

$$-\frac{19}{7}y+\frac{407}{14}+y=14.5$$

Tāpiri -\frac{19y}{7} ki te y.

$$-\frac{12}{7}y+\frac{407}{14}=14.5$$

Me tango \frac{407}{14} mai i ngā taha e rua o te whārite.

$$-\frac{12}{7}y=-\frac{102}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{12}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{17}{2}$$

Whakaurua te \frac{17}{2} mō y ki x=-\frac{19}{7}y+\frac{407}{14}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{19}{7}\times \frac{17}{2}+\frac{407}{14}$$

Whakareatia -\frac{19}{7} ki te \frac{17}{2} 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.

$$x=\frac{-323+407}{14}$$

Tāpiri \frac{407}{14} ki te -\frac{323}{14} 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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=\frac{17}{2}$$

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.

$$70x+190y=2035,x+y=14.5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}70&190\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}70&190\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}70&190\\1&1\end{matrix}\right))\left(\begin{matrix}70&190\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&190\\1&1\end{matrix}\right))\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&190\\1&1\end{matrix}\right))\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}70&190\\1&1\end{matrix}\right))\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{70-190}&-\frac{190}{70-190}\\-\frac{1}{70-190}&\frac{70}{70-190}\end{matrix}\right)\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{120}&\frac{19}{12}\\\frac{1}{120}&-\frac{7}{12}\end{matrix}\right)\left(\begin{matrix}2035\\14.5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{120}\times 2035+\frac{19}{12}\times 14.5\\\frac{1}{120}\times 2035-\frac{7}{12}\times 14.5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\\frac{17}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=\frac{17}{2}$$

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.

$$70x+190y=2035,x+y=14.5$$

Kia ōrite ai a 70x 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 70.

$$70x+190y=2035,70x+70y=70\times 14.5$$

Whakarūnātia.

$$70x+190y=2035,70x+70y=1015$$

Me tango 70x+70y=1015 mai i 70x+190y=2035 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$70x-70x+190y-70y=2035-1015$$

Tāpiri 70x ki te -70x. Ka whakakore atu ngā kupu 70x me -70x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$190y-70y=2035-1015$$

Tāpiri 190y ki te -70y.

$$120y=2035-1015$$

Tāpiri 2035 ki te -1015.

$$120y=1020$$

Whakawehea ngā taha e rua ki te 120.

$$y=\frac{17}{2}$$

Whakaurua te \frac{17}{2} mō y ki x+y=14.5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+\frac{17}{2}=14.5$$

Me tango \frac{17}{2} mai i ngā taha e rua o te whārite.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=\frac{17}{2}$$