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

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

Whakaoti mō x, y

$$x=-1$$

$$y=0$$

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.

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

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

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

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

$$4x=7y-4$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x=\frac{7}{4}y-1$$

Whakakapia te \frac{7y}{4}-1 mō te x ki tērā atu whārite, 7x+5y=-7.

$$7\left(\frac{7}{4}y-1\right)+5y=-7$$

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

$$\frac{49}{4}y-7+5y=-7$$

Tāpiri \frac{49y}{4} ki te 5y.

$$\frac{69}{4}y-7=-7$$

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

$$\frac{69}{4}y=0$$

Whakawehea ngā taha e rua o te whārite ki te \frac{69}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=0$$

Whakaurua te 0 mō y ki x=\frac{7}{4}y-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=0$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-7\\7&5\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{69}&\frac{7}{69}\\-\frac{7}{69}&\frac{4}{69}\end{matrix}\right)\left(\begin{matrix}-4\\-7\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{69}\left(-4\right)+\frac{7}{69}\left(-7\right)\\-\frac{7}{69}\left(-4\right)+\frac{4}{69}\left(-7\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=0$$

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.

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

Kia ōrite ai a 4x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

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

Whakarūnātia.

$$28x-49y=-28,28x+20y=-28$$

Me tango 28x+20y=-28 mai i 28x-49y=-28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x-49y-20y=-28+28$$

Tāpiri 28x ki te -28x. Ka whakakore atu ngā kupu 28x me -28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-49y-20y=-28+28$$

Tāpiri -49y ki te -20y.

$$-69y=-28+28$$

Tāpiri -28 ki te 28.

$$-69y=0$$

Whakawehea ngā taha e rua ki te -69.

$$y=0$$

Whakaurua te 0 mō y ki 7x+5y=-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x=-7$$

Whakawehea ngā taha e rua ki te 7.

$$x=-1$$

Kua oti te pūnaha te whakatau.

$$x=-1,y=0$$