Whakaoti i te {l}{x+y=220}{3/5x=38y-5}

$$\left\{ \begin{array} { l } { x + y = 220 } \\ { \frac { 3 } { 5 } x = 38 y - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{41775}{193} = 216\frac{87}{193} \approx 216.450777202$$

$$y = \frac{685}{193} = 3\frac{106}{193} \approx 3.549222798$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 38y mai i ngā taha e rua.

$$\frac{3}{5}x-38y=-5$$

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.

$$x+y=220,\frac{3}{5}x-38y=-5$$

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{3}{5}x-38y=-5.

$$\frac{3}{5}\left(-y+220\right)-38y=-5$$

Whakareatia \frac{3}{5} ki te -y+220.

$$-\frac{3}{5}y+132-38y=-5$$

Tāpiri -\frac{3y}{5} ki te -38y.

$$-\frac{193}{5}y+132=-5$$

Me tango 132 mai i ngā taha e rua o te whārite.

$$-\frac{193}{5}y=-137$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{193}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{685}{193}$$

Whakaurua te \frac{685}{193} 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.

$$x=-\frac{685}{193}+220$$

Tāpiri 220 ki te -\frac{685}{193}.

$$x=\frac{41775}{193}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{41775}{193},y=\frac{685}{193}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 38y mai i ngā taha e rua.

$$\frac{3}{5}x-38y=-5$$

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

$$x+y=220,\frac{3}{5}x-38y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right))\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\\frac{3}{5}&-38\end{matrix}\right))\left(\begin{matrix}220\\-5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{38}{-38-\frac{3}{5}}&-\frac{1}{-38-\frac{3}{5}}\\-\frac{\frac{3}{5}}{-38-\frac{3}{5}}&\frac{1}{-38-\frac{3}{5}}\end{matrix}\right)\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{190}{193}&\frac{5}{193}\\\frac{3}{193}&-\frac{5}{193}\end{matrix}\right)\left(\begin{matrix}220\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{190}{193}\times 220+\frac{5}{193}\left(-5\right)\\\frac{3}{193}\times 220-\frac{5}{193}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{41775}{193}\\\frac{685}{193}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{41775}{193},y=\frac{685}{193}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 38y mai i ngā taha e rua.

$$\frac{3}{5}x-38y=-5$$

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.

$$x+y=220,\frac{3}{5}x-38y=-5$$

Kia ōrite ai a x me \frac{3x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{3}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$\frac{3}{5}x+\frac{3}{5}y=\frac{3}{5}\times 220,\frac{3}{5}x-38y=-5$$

Whakarūnātia.

$$\frac{3}{5}x+\frac{3}{5}y=132,\frac{3}{5}x-38y=-5$$

Me tango \frac{3}{5}x-38y=-5 mai i \frac{3}{5}x+\frac{3}{5}y=132 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{3}{5}x-\frac{3}{5}x+\frac{3}{5}y+38y=132+5$$

Tāpiri \frac{3x}{5} ki te -\frac{3x}{5}. Ka whakakore atu ngā kupu \frac{3x}{5} me -\frac{3x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{3}{5}y+38y=132+5$$

Tāpiri \frac{3y}{5} ki te 38y.

$$\frac{193}{5}y=132+5$$

Tāpiri 132 ki te 5.

$$\frac{193}{5}y=137$$

Whakawehea ngā taha e rua o te whārite ki te \frac{193}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{685}{193}$$

Whakaurua te \frac{685}{193} mō y ki \frac{3}{5}x-38y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{3}{5}x-38\times \frac{685}{193}=-5$$

Whakareatia -38 ki te \frac{685}{193}.

$$\frac{3}{5}x-\frac{26030}{193}=-5$$

Me tāpiri \frac{26030}{193} ki ngā taha e rua o te whārite.

$$\frac{3}{5}x=\frac{25065}{193}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{41775}{193}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{41775}{193},y=\frac{685}{193}$$