Whakaoti i te {l}{x+y=240}{0.12x+0.06y=240*0.08}

$$\left. \begin{array} { l } { x + y = 240 } \\ { 0.12 x + 0.06 y = 240 \times 0.08 } \end{array} \right.$$

Whakaoti mō x, y

$$x=80$$

$$y=160$$

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.

$$x+y=240,0.12x+0.06y=19.2$$

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

$$x+y=240$$

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

$$x=-y+240$$

Whakakapia te -y+240 mō te x ki tērā atu whārite, 0.12x+0.06y=19.2.

$$0.12\left(-y+240\right)+0.06y=19.2$$

Whakareatia 0.12 ki te -y+240.

$$-0.12y+28.8+0.06y=19.2$$

Tāpiri -\frac{3y}{25} ki te \frac{3y}{50}.

$$-0.06y+28.8=19.2$$

Me tango 28.8 mai i ngā taha e rua o te whārite.

$$-0.06y=-9.6$$

Whakawehea ngā taha e rua o te whārite ki te -0.06, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=160$$

Whakaurua te 160 mō y ki x=-y+240. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-160+240$$

Tāpiri 240 ki te -160.

$$x=80$$

Kua oti te pūnaha te whakatau.

$$x=80,y=160$$

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.

$$x+y=240,0.12x+0.06y=19.2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right))\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right))\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right))\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\0.12&0.06\end{matrix}\right))\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{0.06}{0.06-0.12}&-\frac{1}{0.06-0.12}\\-\frac{0.12}{0.06-0.12}&\frac{1}{0.06-0.12}\end{matrix}\right)\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&\frac{50}{3}\\2&-\frac{50}{3}\end{matrix}\right)\left(\begin{matrix}240\\19.2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-240+\frac{50}{3}\times 19.2\\2\times 240-\frac{50}{3}\times 19.2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80\\160\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=80,y=160$$

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.

$$x+y=240,0.12x+0.06y=19.2$$

Kia ōrite ai a x me \frac{3x}{25}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 0.12 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$0.12x+0.12y=0.12\times 240,0.12x+0.06y=19.2$$

Whakarūnātia.

$$0.12x+0.12y=28.8,0.12x+0.06y=19.2$$

Me tango 0.12x+0.06y=19.2 mai i 0.12x+0.12y=28.8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.12x-0.12x+0.12y-0.06y=\frac{144-96}{5}$$

Tāpiri \frac{3x}{25} ki te -\frac{3x}{25}. Ka whakakore atu ngā kupu \frac{3x}{25} me -\frac{3x}{25}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$0.12y-0.06y=\frac{144-96}{5}$$

Tāpiri \frac{3y}{25} ki te -\frac{3y}{50}.

$$0.06y=\frac{144-96}{5}$$

Tāpiri 28.8 ki te -19.2 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.

$$0.06y=9.6$$

Whakawehea ngā taha e rua o te whārite ki te 0.06, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=160$$

Whakaurua te 160 mō y ki 0.12x+0.06y=19.2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$0.12x+0.06\times 160=19.2$$

Whakareatia 0.06 ki te 160.

$$0.12x+9.6=19.2$$

Me tango 9.6 mai i ngā taha e rua o te whārite.

$$0.12x=9.6$$

Whakawehea ngā taha e rua o te whārite ki te 0.12, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=80$$

Kua oti te pūnaha te whakatau.

$$x=80,y=160$$