Whakaoti i te {l}{-7x-4y=62}{3x+y=-2}

$$\left\{ \begin{array} { l } { - 7 x - 4 y = 62 } \\ { 3 x + y = - 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{54}{5} = 10\frac{4}{5} = 10.8$$

$$y = -\frac{172}{5} = -34\frac{2}{5} = -34.4$$

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.

$$-7x-4y=62,3x+y=-2$$

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

$$-7x-4y=62$$

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

$$-7x=4y+62$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(4y+62\right)$$

Whakareatia -\frac{1}{7} ki te 4y+62.

$$x=-\frac{4}{7}y-\frac{62}{7}$$

Whakakapia te \frac{-4y-62}{7} mō te x ki tērā atu whārite, 3x+y=-2.

$$3\left(-\frac{4}{7}y-\frac{62}{7}\right)+y=-2$$

Whakareatia 3 ki te \frac{-4y-62}{7}.

$$-\frac{12}{7}y-\frac{186}{7}+y=-2$$

Tāpiri -\frac{12y}{7} ki te y.

$$-\frac{5}{7}y-\frac{186}{7}=-2$$

Me tāpiri \frac{186}{7} ki ngā taha e rua o te whārite.

$$-\frac{5}{7}y=\frac{172}{7}$$

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

$$y=-\frac{172}{5}$$

Whakaurua te -\frac{172}{5} mō y ki x=-\frac{4}{7}y-\frac{62}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{4}{7}\left(-\frac{172}{5}\right)-\frac{62}{7}$$

Whakareatia -\frac{4}{7} ki te -\frac{172}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{688}{35}-\frac{62}{7}$$

Tāpiri -\frac{62}{7} ki te \frac{688}{35} 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.

$$x=\frac{54}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{54}{5},y=-\frac{172}{5}$$

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.

$$-7x-4y=62,3x+y=-2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}62\\-2\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&-4\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right))\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right))\left(\begin{matrix}62\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right))\left(\begin{matrix}62\\-2\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&-4\\3&1\end{matrix}\right))\left(\begin{matrix}62\\-2\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-7-\left(-4\times 3\right)}&-\frac{-4}{-7-\left(-4\times 3\right)}\\-\frac{3}{-7-\left(-4\times 3\right)}&-\frac{7}{-7-\left(-4\times 3\right)}\end{matrix}\right)\left(\begin{matrix}62\\-2\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{4}{5}\\-\frac{3}{5}&-\frac{7}{5}\end{matrix}\right)\left(\begin{matrix}62\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 62+\frac{4}{5}\left(-2\right)\\-\frac{3}{5}\times 62-\frac{7}{5}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{54}{5}\\-\frac{172}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{54}{5},y=-\frac{172}{5}$$

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.

$$-7x-4y=62,3x+y=-2$$

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.

$$3\left(-7\right)x+3\left(-4\right)y=3\times 62,-7\times 3x-7y=-7\left(-2\right)$$

Whakarūnātia.

$$-21x-12y=186,-21x-7y=14$$

Me tango -21x-7y=14 mai i -21x-12y=186 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-21x+21x-12y+7y=186-14$$

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.

$$-12y+7y=186-14$$

Tāpiri -12y ki te 7y.

$$-5y=186-14$$

Tāpiri 186 ki te -14.

$$-5y=172$$

Whakawehea ngā taha e rua ki te -5.

$$y=-\frac{172}{5}$$

Whakaurua te -\frac{172}{5} mō y ki 3x+y=-2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-\frac{172}{5}=-2$$

Me tāpiri \frac{172}{5} ki ngā taha e rua o te whārite.

$$3x=\frac{162}{5}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{54}{5}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{54}{5},y=-\frac{172}{5}$$