Whakaoti i te {l}{2x+5y=16}{3x-7y=24}

$$\left. \begin{array} { l } { 2 x + 5 y = 16 } \\ { 3 x - 7 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=0$$

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.

$$2x+5y=16,3x-7y=24$$

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

$$2x+5y=16$$

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

$$2x=-5y+16$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y+16\right)$$

Whakareatia \frac{1}{2} ki te -5y+16.

$$x=-\frac{5}{2}y+8$$

Whakakapia te -\frac{5y}{2}+8 mō te x ki tērā atu whārite, 3x-7y=24.

$$3\left(-\frac{5}{2}y+8\right)-7y=24$$

Whakareatia 3 ki te -\frac{5y}{2}+8.

$$-\frac{15}{2}y+24-7y=24$$

Tāpiri -\frac{15y}{2} ki te -7y.

$$-\frac{29}{2}y+24=24$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$-\frac{29}{2}y=0$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{5}{2}y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=0$$

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.

$$2x+5y=16,3x-7y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\3&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}16\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\3&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\3&-7\end{matrix}\right))\left(\begin{matrix}2&5\\3&-7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-7\end{matrix}\right))\left(\begin{matrix}16\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-7\end{matrix}\right))\left(\begin{matrix}16\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\3&-7\end{matrix}\right))\left(\begin{matrix}16\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{2\left(-7\right)-5\times 3}&-\frac{5}{2\left(-7\right)-5\times 3}\\-\frac{3}{2\left(-7\right)-5\times 3}&\frac{2}{2\left(-7\right)-5\times 3}\end{matrix}\right)\left(\begin{matrix}16\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{29}&\frac{5}{29}\\\frac{3}{29}&-\frac{2}{29}\end{matrix}\right)\left(\begin{matrix}16\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{29}\times 16+\frac{5}{29}\times 24\\\frac{3}{29}\times 16-\frac{2}{29}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=0$$

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.

$$2x+5y=16,3x-7y=24$$

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.

$$3\times 2x+3\times 5y=3\times 16,2\times 3x+2\left(-7\right)y=2\times 24$$

Whakarūnātia.

$$6x+15y=48,6x-14y=48$$

Me tango 6x-14y=48 mai i 6x+15y=48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x+15y+14y=48-48$$

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.

$$15y+14y=48-48$$

Tāpiri 15y ki te 14y.

$$29y=48-48$$

Tāpiri 48 ki te -48.

$$29y=0$$

Whakawehea ngā taha e rua ki te 29.

$$y=0$$

Whakaurua te 0 mō y ki 3x-7y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x=24$$

Whakawehea ngā taha e rua ki te 3.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=0$$