Whakaoti i te {l}{x+1.5y=12}{x+3.5y=16}

$$\left. \begin{array} { l } { x + 1.5 y = 12 } \\ { x + 3.5 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x=9$$

$$y=2$$

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.

$$x+1.5y=12,x+3.5y=16$$

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

$$x+1.5y=12$$

Me tango \frac{3y}{2} mai i ngā taha e rua o te whārite.

$$x=-1.5y+12$$

Whakakapia te -\frac{3y}{2}+12 mō te x ki tērā atu whārite, x+3.5y=16.

$$-1.5y+12+3.5y=16$$

Tāpiri -\frac{3y}{2} ki te \frac{7y}{2}.

$$2y+12=16$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$2y=4$$

Whakawehea ngā taha e rua ki te 2.

$$y=2$$

Whakaurua te 2 mō y ki x=-1.5y+12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-1.5\times 2+12$$

Whakareatia -1.5 ki te 2.

$$x=-3+12$$

Tāpiri 12 ki te -3.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=2$$

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.

$$x+1.5y=12,x+3.5y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right))\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1.5\\1&3.5\end{matrix}\right))\left(\begin{matrix}12\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3.5}{3.5-1.5}&-\frac{1.5}{3.5-1.5}\\-\frac{1}{3.5-1.5}&\frac{1}{3.5-1.5}\end{matrix}\right)\left(\begin{matrix}12\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4}&-\frac{3}{4}\\-\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}12\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4}\times 12-\frac{3}{4}\times 16\\-\frac{1}{2}\times 12+\frac{1}{2}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=9,y=2$$

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.

$$x+1.5y=12,x+3.5y=16$$

Me tango x+3.5y=16 mai i x+1.5y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+1.5y-3.5y=12-16$$

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

$$1.5y-3.5y=12-16$$

Tāpiri \frac{3y}{2} ki te -\frac{7y}{2}.

$$-2y=12-16$$

Tāpiri 12 ki te -16.

$$-2y=-4$$

Whakawehea ngā taha e rua ki te -2.

$$y=2$$

Whakaurua te 2 mō y ki x+3.5y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+3.5\times 2=16$$

Whakareatia 3.5 ki te 2.

$$x+7=16$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=9$$

Kua oti te pūnaha te whakatau.

$$x=9,y=2$$