Whakaoti i te {l}{0.04x+0.02y=5}{0.5(x-2)-0.4y=29}

$$\left. \begin{array} { l } { 0.04 x + 0.02 y = 5 } \\ { 0.5 ( x - 2 ) - 0.4 y = 29 } \end{array} \right.$$

Whakaoti mō x, y

$$x=100$$

$$y=50$$

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.

$$0.04x+0.02y=5,0.5\left(x-2\right)-0.4y=29$$

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

$$0.04x+0.02y=5$$

Me tango \frac{y}{50} mai i ngā taha e rua o te whārite.

$$0.04x=-0.02y+5$$

Me whakarea ngā taha e rua ki te 25.

$$x=25\left(-0.02y+5\right)$$

Whakareatia 25 ki te -\frac{y}{50}+5.

$$x=-0.5y+125$$

Whakakapia te -\frac{y}{2}+125 mō te x ki tērā atu whārite, 0.5\left(x-2\right)-0.4y=29.

$$0.5\left(-0.5y+125-2\right)-0.4y=29$$

Tāpiri 125 ki te -2.

$$0.5\left(-0.5y+123\right)-0.4y=29$$

Whakareatia 0.5 ki te -\frac{y}{2}+123.

$$-0.25y+61.5-0.4y=29$$

Tāpiri -\frac{y}{4} ki te -\frac{2y}{5}.

$$-0.65y+61.5=29$$

Me tango 61.5 mai i ngā taha e rua o te whārite.

$$-0.65y=-32.5$$

Whakawehea ngā taha e rua o te whārite ki te -0.65, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=50$$

Whakaurua te 50 mō y ki x=-0.5y+125. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-0.5\times 50+125$$

Whakareatia -0.5 ki te 50.

$$x=-25+125$$

Tāpiri 125 ki te -25.

$$x=100$$

Kua oti te pūnaha te whakatau.

$$x=100,y=50$$

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.

$$0.04x+0.02y=5,0.5\left(x-2\right)-0.4y=29$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$0.5\left(x-2\right)-0.4y=29$$

Whakareatia 0.5 ki te x-2.

$$0.5x-1-0.4y=29$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$0.5x-0.4y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.04&0.02\\0.5&-0.4\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{0.4}{0.04\left(-0.4\right)-0.02\times 0.5}&-\frac{0.02}{0.04\left(-0.4\right)-0.02\times 0.5}\\-\frac{0.5}{0.04\left(-0.4\right)-0.02\times 0.5}&\frac{0.04}{0.04\left(-0.4\right)-0.02\times 0.5}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{13}&\frac{10}{13}\\\frac{250}{13}&-\frac{20}{13}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{13}\times 5+\frac{10}{13}\times 30\\\frac{250}{13}\times 5-\frac{20}{13}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\50\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=100,y=50$$