Whakaoti i te {l}{x+y=220}{2/5x=3/8y-5}

$$\left\{ \begin{array} { l } { x + y = 220 } \\ { \frac { 2 } { 5 } x = \frac { 3 } { 8 } y - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=100$$

$$y=120$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{8}y mai i ngā taha e rua.

$$\frac{2}{5}x-\frac{3}{8}y=-5$$

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.

$$x+y=220,\frac{2}{5}x-\frac{3}{8}y=-5$$

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

$$x+y=220$$

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

$$x=-y+220$$

Whakakapia te -y+220 mō te x ki tērā atu whārite, \frac{2}{5}x-\frac{3}{8}y=-5.

$$\frac{2}{5}\left(-y+220\right)-\frac{3}{8}y=-5$$

Whakareatia \frac{2}{5} ki te -y+220.

$$-\frac{2}{5}y+88-\frac{3}{8}y=-5$$

Tāpiri -\frac{2y}{5} ki te -\frac{3y}{8}.

$$-\frac{31}{40}y+88=-5$$

Me tango 88 mai i ngā taha e rua o te whārite.

$$-\frac{31}{40}y=-93$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{31}{40}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=120$$

Whakaurua te 120 mō y ki x=-y+220. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-120+220$$

Tāpiri 220 ki te -120.

$$x=100$$

Kua oti te pūnaha te whakatau.

$$x=100,y=120$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{8}y mai i ngā taha e rua.

$$\frac{2}{5}x-\frac{3}{8}y=-5$$

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

$$x+y=220,\frac{2}{5}x-\frac{3}{8}y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right))\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{2}{5}&-\frac{3}{8}\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{3}{8}}{-\frac{3}{8}-\frac{2}{5}}&-\frac{1}{-\frac{3}{8}-\frac{2}{5}}\\-\frac{\frac{2}{5}}{-\frac{3}{8}-\frac{2}{5}}&\frac{1}{-\frac{3}{8}-\frac{2}{5}}\end{matrix}\right)\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{31}&\frac{40}{31}\\\frac{16}{31}&-\frac{40}{31}\end{matrix}\right)\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{31}\times 220+\frac{40}{31}\left(-5\right)\\\frac{16}{31}\times 220-\frac{40}{31}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}100\\120\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=100,y=120$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te \frac{3}{8}y mai i ngā taha e rua.

$$\frac{2}{5}x-\frac{3}{8}y=-5$$

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.

$$x+y=220,\frac{2}{5}x-\frac{3}{8}y=-5$$

Kia ōrite ai a x me \frac{2x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{2}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$\frac{2}{5}x+\frac{2}{5}y=\frac{2}{5}\times 220,\frac{2}{5}x-\frac{3}{8}y=-5$$

Whakarūnātia.

$$\frac{2}{5}x+\frac{2}{5}y=88,\frac{2}{5}x-\frac{3}{8}y=-5$$

Me tango \frac{2}{5}x-\frac{3}{8}y=-5 mai i \frac{2}{5}x+\frac{2}{5}y=88 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{2}{5}x-\frac{2}{5}x+\frac{2}{5}y+\frac{3}{8}y=88+5$$

Tāpiri \frac{2x}{5} ki te -\frac{2x}{5}. Ka whakakore atu ngā kupu \frac{2x}{5} me -\frac{2x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{2}{5}y+\frac{3}{8}y=88+5$$

Tāpiri \frac{2y}{5} ki te \frac{3y}{8}.

$$\frac{31}{40}y=88+5$$

Tāpiri 88 ki te 5.

$$\frac{31}{40}y=93$$

Whakawehea ngā taha e rua o te whārite ki te \frac{31}{40}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=120$$

Whakaurua te 120 mō y ki \frac{2}{5}x-\frac{3}{8}y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{2}{5}x-\frac{3}{8}\times 120=-5$$

Whakareatia -\frac{3}{8} ki te 120.

$$\frac{2}{5}x-45=-5$$

Me tāpiri 45 ki ngā taha e rua o te whārite.

$$\frac{2}{5}x=40$$

Whakawehea ngā taha e rua o te whārite ki te \frac{2}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=100$$

Kua oti te pūnaha te whakatau.

$$x=100,y=120$$