Whakaoti i te {l}{45+.25x=y}{35+.30x=y}

$$\left. \begin{array} { l } { 45 + .25 x = y } \\ { 35 + .30 x = y } \end{array} \right.$$

Whakaoti mō x, y

$$x=200$$

$$y=95$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$45+0.25x-y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.25x-y=-45$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$35+0.3x-y=0$$

Tangohia te 35 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.3x-y=-35$$

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.

$$0.25x-y=-45,0.3x-y=-35$$

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

$$0.25x-y=-45$$

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

$$0.25x=y-45$$

Me whakarea ngā taha e rua ki te 4.

$$x=4\left(y-45\right)$$

Whakareatia 4 ki te y-45.

$$x=4y-180$$

Whakakapia te -180+4y mō te x ki tērā atu whārite, 0.3x-y=-35.

$$0.3\left(4y-180\right)-y=-35$$

Whakareatia 0.3 ki te -180+4y.

$$1.2y-54-y=-35$$

Tāpiri \frac{6y}{5} ki te -y.

$$0.2y-54=-35$$

Me tāpiri 54 ki ngā taha e rua o te whārite.

$$0.2y=19$$

Me whakarea ngā taha e rua ki te 5.

$$y=95$$

Whakaurua te 95 mō y ki x=4y-180. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4\times 95-180$$

Whakareatia 4 ki te 95.

$$x=380-180$$

Tāpiri -180 ki te 380.

$$x=200$$

Kua oti te pūnaha te whakatau.

$$x=200,y=95$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$45+0.25x-y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.25x-y=-45$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$35+0.3x-y=0$$

Tangohia te 35 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.3x-y=-35$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$0.25x-y=-45,0.3x-y=-35$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right))\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right))\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right))\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}0.25&-1\\0.3&-1\end{matrix}\right))\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{0.25\left(-1\right)-\left(-0.3\right)}&-\frac{-1}{0.25\left(-1\right)-\left(-0.3\right)}\\-\frac{0.3}{0.25\left(-1\right)-\left(-0.3\right)}&\frac{0.25}{0.25\left(-1\right)-\left(-0.3\right)}\end{matrix}\right)\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20&20\\-6&5\end{matrix}\right)\left(\begin{matrix}-45\\-35\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-20\left(-45\right)+20\left(-35\right)\\-6\left(-45\right)+5\left(-35\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\95\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=200,y=95$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$45+0.25x-y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.25x-y=-45$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$35+0.3x-y=0$$

Tangohia te 35 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$0.3x-y=-35$$

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.

$$0.25x-y=-45,0.3x-y=-35$$

Me tango 0.3x-y=-35 mai i 0.25x-y=-45 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$0.25x-0.3x-y+y=-45+35$$

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

$$0.25x-0.3x=-45+35$$

Tāpiri \frac{x}{4} ki te -\frac{3x}{10}.

$$-0.05x=-45+35$$

Tāpiri -45 ki te 35.

$$-0.05x=-10$$

Me whakarea ngā taha e rua ki te -20.

$$x=200$$

Whakaurua te 200 mō x ki 0.3x-y=-35. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$0.3\times 200-y=-35$$

Whakareatia 0.3 ki te 200.

$$60-y=-35$$

Me tango 60 mai i ngā taha e rua o te whārite.

$$-y=-95$$

Whakawehea ngā taha e rua ki te -1.

$$y=95$$

Kua oti te pūnaha te whakatau.

$$x=200,y=95$$