Whakaoti i te {l}{2x+y=45}{3x+5y=70}

$$\left. \begin{array} { l } { 2 x + y = 45 } \\ { 3 x + 5 y = 70 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{155}{7} = 22\frac{1}{7} \approx 22.142857143$$

$$y=\frac{5}{7}\approx 0.714285714$$

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.

$$2x+y=45,3x+5y=70$$

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

$$2x+y=45$$

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

$$2x=-y+45$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-y+45\right)$$

Whakareatia \frac{1}{2} ki te -y+45.

$$x=-\frac{1}{2}y+\frac{45}{2}$$

Whakakapia te \frac{-y+45}{2} mō te x ki tērā atu whārite, 3x+5y=70.

$$3\left(-\frac{1}{2}y+\frac{45}{2}\right)+5y=70$$

Whakareatia 3 ki te \frac{-y+45}{2}.

$$-\frac{3}{2}y+\frac{135}{2}+5y=70$$

Tāpiri -\frac{3y}{2} ki te 5y.

$$\frac{7}{2}y+\frac{135}{2}=70$$

Me tango \frac{135}{2} mai i ngā taha e rua o te whārite.

$$\frac{7}{2}y=\frac{5}{2}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{5}{7}$$

Whakaurua te \frac{5}{7} mō y ki x=-\frac{1}{2}y+\frac{45}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{2}\times \frac{5}{7}+\frac{45}{2}$$

Whakareatia -\frac{1}{2} ki te \frac{5}{7} 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.

$$x=-\frac{5}{14}+\frac{45}{2}$$

Tāpiri \frac{45}{2} ki te -\frac{5}{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.

$$x=\frac{155}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{155}{7},y=\frac{5}{7}$$

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.

$$2x+y=45,3x+5y=70$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\70\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&1\\3&5\end{matrix}\right))\left(\begin{matrix}2&1\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&5\end{matrix}\right))\left(\begin{matrix}45\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&5\end{matrix}\right))\left(\begin{matrix}45\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&1\\3&5\end{matrix}\right))\left(\begin{matrix}45\\70\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-3}&-\frac{1}{2\times 5-3}\\-\frac{3}{2\times 5-3}&\frac{2}{2\times 5-3}\end{matrix}\right)\left(\begin{matrix}45\\70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}&-\frac{1}{7}\\-\frac{3}{7}&\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}45\\70\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}\times 45-\frac{1}{7}\times 70\\-\frac{3}{7}\times 45+\frac{2}{7}\times 70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{155}{7}\\\frac{5}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{155}{7},y=\frac{5}{7}$$

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.

$$2x+y=45,3x+5y=70$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3y=3\times 45,2\times 3x+2\times 5y=2\times 70$$

Whakarūnātia.

$$6x+3y=135,6x+10y=140$$

Me tango 6x+10y=140 mai i 6x+3y=135 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+3y-10y=135-140$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y-10y=135-140$$

Tāpiri 3y ki te -10y.

$$-7y=135-140$$

Tāpiri 135 ki te -140.

$$-7y=-5$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{5}{7}$$

Whakaurua te \frac{5}{7} mō y ki 3x+5y=70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+5\times \frac{5}{7}=70$$

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

$$3x+\frac{25}{7}=70$$

Me tango \frac{25}{7} mai i ngā taha e rua o te whārite.

$$3x=\frac{465}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{155}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{155}{7},y=\frac{5}{7}$$