Whakaoti i te {l}{2x+5y=-10}{y=-7/5x+3}

$$\left. \begin{array} { l } { 2 x + 5 y = - 10 } \\ { y = - \frac { 7 } { 5 } x + 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te \frac{7}{5}x ki ngā taha e rua.

$$y+\frac{7}{5}x=3$$

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.

$$2x+5y=-10,\frac{7}{5}x+y=3$$

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

$$2x+5y=-10$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$2x=-5y-10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-5y-10\right)$$

Whakareatia \frac{1}{2} ki te -5y-10.

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

Whakakapia te -\frac{5y}{2}-5 mō te x ki tērā atu whārite, \frac{7}{5}x+y=3.

$$\frac{7}{5}\left(-\frac{5}{2}y-5\right)+y=3$$

Whakareatia \frac{7}{5} ki te -\frac{5y}{2}-5.

$$-\frac{7}{2}y-7+y=3$$

Tāpiri -\frac{7y}{2} ki te y.

$$-\frac{5}{2}y-7=3$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$-\frac{5}{2}y=10$$

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

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{5}{2}y-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{2}\left(-4\right)-5$$

Whakareatia -\frac{5}{2} ki te -4.

$$x=10-5$$

Tāpiri -5 ki te 10.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te \frac{7}{5}x ki ngā taha e rua.

$$y+\frac{7}{5}x=3$$

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

$$2x+5y=-10,\frac{7}{5}x+y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right))\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right))\left(\begin{matrix}-10\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right))\left(\begin{matrix}-10\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&5\\\frac{7}{5}&1\end{matrix}\right))\left(\begin{matrix}-10\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2-5\times \frac{7}{5}}&-\frac{5}{2-5\times \frac{7}{5}}\\-\frac{\frac{7}{5}}{2-5\times \frac{7}{5}}&\frac{2}{2-5\times \frac{7}{5}}\end{matrix}\right)\left(\begin{matrix}-10\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&1\\\frac{7}{25}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}-10\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\left(-10\right)+3\\\frac{7}{25}\left(-10\right)-\frac{2}{5}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-4$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te \frac{7}{5}x ki ngā taha e rua.

$$y+\frac{7}{5}x=3$$

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.

$$2x+5y=-10,\frac{7}{5}x+y=3$$

Kia ōrite ai a 2x me \frac{7x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{7}{5} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$\frac{7}{5}\times 2x+\frac{7}{5}\times 5y=\frac{7}{5}\left(-10\right),2\times \frac{7}{5}x+2y=2\times 3$$

Whakarūnātia.

$$\frac{14}{5}x+7y=-14,\frac{14}{5}x+2y=6$$

Me tango \frac{14}{5}x+2y=6 mai i \frac{14}{5}x+7y=-14 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$\frac{14}{5}x-\frac{14}{5}x+7y-2y=-14-6$$

Tāpiri \frac{14x}{5} ki te -\frac{14x}{5}. Ka whakakore atu ngā kupu \frac{14x}{5} me -\frac{14x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y-2y=-14-6$$

Tāpiri 7y ki te -2y.

$$5y=-14-6$$

Tāpiri -14 ki te -6.

$$5y=-20$$

Whakawehea ngā taha e rua ki te 5.

$$y=-4$$

Whakaurua te -4 mō y ki \frac{7}{5}x+y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{7}{5}x-4=3$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$\frac{7}{5}x=7$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-4$$