Whakaoti i te {l}{7x+y=-9}{-3x-y=5}

$$\left. \begin{array} { l } { 7 x + y = - 9 } \\ { - 3 x - y = 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-2$$

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.

$$7x+y=-9,-3x-y=5$$

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

$$7x+y=-9$$

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

$$7x=-y-9$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-y-9\right)$$

Whakareatia \frac{1}{7} ki te -y-9.

$$x=-\frac{1}{7}y-\frac{9}{7}$$

Whakakapia te \frac{-y-9}{7} mō te x ki tērā atu whārite, -3x-y=5.

$$-3\left(-\frac{1}{7}y-\frac{9}{7}\right)-y=5$$

Whakareatia -3 ki te \frac{-y-9}{7}.

$$\frac{3}{7}y+\frac{27}{7}-y=5$$

Tāpiri \frac{3y}{7} ki te -y.

$$-\frac{4}{7}y+\frac{27}{7}=5$$

Me tango \frac{27}{7} mai i ngā taha e rua o te whārite.

$$-\frac{4}{7}y=\frac{8}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{1}{7}y-\frac{9}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{7}\left(-2\right)-\frac{9}{7}$$

Whakareatia -\frac{1}{7} ki te -2.

$$x=\frac{2-9}{7}$$

Tāpiri -\frac{9}{7} ki te \frac{2}{7} 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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-2$$

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.

$$7x+y=-9,-3x-y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&1\\-3&-1\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&1\\-3&-1\end{matrix}\right))\left(\begin{matrix}-9\\5\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-2$$

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.

$$7x+y=-9,-3x-y=5$$

Kia ōrite ai a 7x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$-3\times 7x-3y=-3\left(-9\right),7\left(-3\right)x+7\left(-1\right)y=7\times 5$$

Whakarūnātia.

$$-21x-3y=27,-21x-7y=35$$

Me tango -21x-7y=35 mai i -21x-3y=27 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-21x+21x-3y+7y=27-35$$

Tāpiri -21x ki te 21x. Ka whakakore atu ngā kupu -21x me 21x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y+7y=27-35$$

Tāpiri -3y ki te 7y.

$$4y=27-35$$

Tāpiri 27 ki te -35.

$$4y=-8$$

Whakawehea ngā taha e rua ki te 4.

$$y=-2$$

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

$$-3x-\left(-2\right)=5$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$-3x=3$$

Whakawehea ngā taha e rua ki te -3.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=-2$$