Whakaoti i te {l}{-4x+3y=-5}{-7x+3y=-20}

$$\left. \begin{array} { l } { - 4 x + 3 y = - 5 } \\ { - 7 x + 3 y = - 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=5$$

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.

$$-4x+3y=-5,-7x+3y=-20$$

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

$$-4x+3y=-5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$-4x=-3y-5$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(-3y-5\right)$$

Whakareatia -\frac{1}{4} ki te -3y-5.

$$x=\frac{3}{4}y+\frac{5}{4}$$

Whakakapia te \frac{3y+5}{4} mō te x ki tērā atu whārite, -7x+3y=-20.

$$-7\left(\frac{3}{4}y+\frac{5}{4}\right)+3y=-20$$

Whakareatia -7 ki te \frac{3y+5}{4}.

$$-\frac{21}{4}y-\frac{35}{4}+3y=-20$$

Tāpiri -\frac{21y}{4} ki te 3y.

$$-\frac{9}{4}y-\frac{35}{4}=-20$$

Me tāpiri \frac{35}{4} ki ngā taha e rua o te whārite.

$$-\frac{9}{4}y=-\frac{45}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

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

$$x=\frac{3}{4}\times 5+\frac{5}{4}$$

Whakareatia \frac{3}{4} ki te 5.

$$x=\frac{15+5}{4}$$

Tāpiri \frac{5}{4} ki te \frac{15}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=5$$

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.

$$-4x+3y=-5,-7x+3y=-20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&3\\-7&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right))\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right))\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right))\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&3\\-7&3\end{matrix}\right))\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-4\times 3-3\left(-7\right)}&-\frac{3}{-4\times 3-3\left(-7\right)}\\-\frac{-7}{-4\times 3-3\left(-7\right)}&-\frac{4}{-4\times 3-3\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\\frac{7}{9}&-\frac{4}{9}\end{matrix}\right)\left(\begin{matrix}-5\\-20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\left(-5\right)-\frac{1}{3}\left(-20\right)\\\frac{7}{9}\left(-5\right)-\frac{4}{9}\left(-20\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=5$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$-4x+3y=-5,-7x+3y=-20$$

Me tango -7x+3y=-20 mai i -4x+3y=-5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+7x+3y-3y=-5+20$$

Tāpiri 3y ki te -3y. Ka whakakore atu ngā kupu 3y me -3y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4x+7x=-5+20$$

Tāpiri -4x ki te 7x.

$$3x=-5+20$$

Tāpiri -5 ki te 20.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Whakaurua te 5 mō x ki -7x+3y=-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-7\times 5+3y=-20$$

Whakareatia -7 ki te 5.

$$-35+3y=-20$$

Me tāpiri 35 ki ngā taha e rua o te whārite.

$$3y=15$$

Whakawehea ngā taha e rua ki te 3.

$$y=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=5$$