Whakaoti i te {l}{x+y=200}{x+1/2y=160}

$$\left. \begin{array} { l } { x + y = 200 } \\ { x + \frac { 1 } { 2 } y = 160 } \end{array} \right.$$

Whakaoti mō x, y

$$x=120$$

$$y=80$$

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.

$$x+y=200,x+\frac{1}{2}y=160$$

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

$$x+y=200$$

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

$$x=-y+200$$

Whakakapia te -y+200 mō te x ki tērā atu whārite, x+\frac{1}{2}y=160.

$$-y+200+\frac{1}{2}y=160$$

Tāpiri -y ki te \frac{y}{2}.

$$-\frac{1}{2}y+200=160$$

Me tango 200 mai i ngā taha e rua o te whārite.

$$-\frac{1}{2}y=-40$$

Me whakarea ngā taha e rua ki te -2.

$$y=80$$

Whakaurua te 80 mō y ki x=-y+200. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-80+200$$

Tāpiri 200 ki te -80.

$$x=120$$

Kua oti te pūnaha te whakatau.

$$x=120,y=80$$

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.

$$x+y=200,x+\frac{1}{2}y=160$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\160\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}200\\160\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}200\\160\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&\frac{1}{2}\end{matrix}\right))\left(\begin{matrix}200\\160\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{1}{2}}{\frac{1}{2}-1}&-\frac{1}{\frac{1}{2}-1}\\-\frac{1}{\frac{1}{2}-1}&\frac{1}{\frac{1}{2}-1}\end{matrix}\right)\left(\begin{matrix}200\\160\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&2\\2&-2\end{matrix}\right)\left(\begin{matrix}200\\160\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-200+2\times 160\\2\times 200-2\times 160\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}120\\80\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=120,y=80$$

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.

$$x+y=200,x+\frac{1}{2}y=160$$

Me tango x+\frac{1}{2}y=160 mai i x+y=200 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-\frac{1}{2}y=200-160$$

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.

$$y-\frac{1}{2}y=200-160$$

Tāpiri y ki te -\frac{y}{2}.

$$\frac{1}{2}y=200-160$$

Tāpiri 200 ki te -160.

$$\frac{1}{2}y=40$$

Me whakarea ngā taha e rua ki te 2.

$$y=80$$

Whakaurua te 80 mō y ki x+\frac{1}{2}y=160. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x+\frac{1}{2}\times 80=160$$

Whakareatia \frac{1}{2} ki te 80.

$$x+40=160$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$x=120$$

Kua oti te pūnaha te whakatau.

$$x=120,y=80$$