Whakaoti i te {l}{4x+3y=-7}{3x-5y=2}

$$\left\{ \begin{array} { l } { 4 x + 3 y = - 7 } \\ { 3 x - 5 y = 2 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-1$$

$$y=-1$$

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.

$$4x+3y=-7,3x-5y=2$$

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

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

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

$$4x=-3y-7$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y-7\right)$$

Whakareatia \frac{1}{4} ki te -3y-7.

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

Whakakapia te \frac{-3y-7}{4} mō te x ki tērā atu whārite, 3x-5y=2.

$$3\left(-\frac{3}{4}y-\frac{7}{4}\right)-5y=2$$

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

$$-\frac{9}{4}y-\frac{21}{4}-5y=2$$

Tāpiri -\frac{9y}{4} ki te -5y.

$$-\frac{29}{4}y-\frac{21}{4}=2$$

Me tāpiri \frac{21}{4} ki ngā taha e rua o te whārite.

$$-\frac{29}{4}y=\frac{29}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{29}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-1$$

Whakaurua te -1 mō y ki x=-\frac{3}{4}y-\frac{7}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\left(-1\right)-\frac{7}{4}$$

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

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

Tāpiri -\frac{7}{4} ki te \frac{3}{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.

$$x=-1$$

Kua oti te pūnaha te whakatau.

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

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.

$$4x+3y=-7,3x-5y=2$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&3\\3&-5\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\3&-5\end{matrix}\right))\left(\begin{matrix}-7\\2\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{29}&\frac{3}{29}\\\frac{3}{29}&-\frac{4}{29}\end{matrix}\right)\left(\begin{matrix}-7\\2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{29}\left(-7\right)+\frac{3}{29}\times 2\\\frac{3}{29}\left(-7\right)-\frac{4}{29}\times 2\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

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.

$$4x+3y=-7,3x-5y=2$$

Kia ōrite ai a 4x 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 4.

$$3\times 4x+3\times 3y=3\left(-7\right),4\times 3x+4\left(-5\right)y=4\times 2$$

Whakarūnātia.

$$12x+9y=-21,12x-20y=8$$

Me tango 12x-20y=8 mai i 12x+9y=-21 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+9y+20y=-21-8$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y+20y=-21-8$$

Tāpiri 9y ki te 20y.

$$29y=-21-8$$

Tāpiri -21 ki te -8.

$$29y=-29$$

Whakawehea ngā taha e rua ki te 29.

$$y=-1$$

Whakaurua te -1 mō y ki 3x-5y=2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-5\left(-1\right)=2$$

Whakareatia -5 ki te -1.

$$3x+5=2$$

Me tango 5 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.

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