Whakaoti i te {c}{7x+5y=5}{-7x+12y=12}

$$\left. \begin{array} { c } { 7 x + 5 y = 5 } \\ { - 7 x + 12 y = 12 } \end{array} \right.$$

Whakaoti mō x, y

$$x=0$$

$$y=1$$

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.

$$7x+5y=5,-7x+12y=12$$

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

$$7x+5y=5$$

Me tango 5y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 7.

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

Whakareatia \frac{1}{7} ki te -5y+5.

$$x=-\frac{5}{7}y+\frac{5}{7}$$

Whakakapia te \frac{-5y+5}{7} mō te x ki tērā atu whārite, -7x+12y=12.

$$-7\left(-\frac{5}{7}y+\frac{5}{7}\right)+12y=12$$

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

$$5y-5+12y=12$$

Tāpiri 5y ki te 12y.

$$17y-5=12$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$17y=17$$

Whakawehea ngā taha e rua ki te 17.

$$y=1$$

Whakaurua te 1 mō y ki x=-\frac{5}{7}y+\frac{5}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{-5+5}{7}$$

Tāpiri \frac{5}{7} ki te -\frac{5}{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.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=1$$

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.

$$7x+5y=5,-7x+12y=12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&5\\-7&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&5\\-7&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&5\\-7&12\end{matrix}\right))\left(\begin{matrix}7&5\\-7&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-7&12\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-7&12\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&5\\-7&12\end{matrix}\right))\left(\begin{matrix}5\\12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{7\times 12-5\left(-7\right)}&-\frac{5}{7\times 12-5\left(-7\right)}\\-\frac{-7}{7\times 12-5\left(-7\right)}&\frac{7}{7\times 12-5\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}5\\12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{119}&-\frac{5}{119}\\\frac{1}{17}&\frac{1}{17}\end{matrix}\right)\left(\begin{matrix}5\\12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{119}\times 5-\frac{5}{119}\times 12\\\frac{1}{17}\times 5+\frac{1}{17}\times 12\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=0,y=1$$

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.

$$7x+5y=5,-7x+12y=12$$

Kia ōrite ai a 7x 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 7.

$$-7\times 7x-7\times 5y=-7\times 5,7\left(-7\right)x+7\times 12y=7\times 12$$

Whakarūnātia.

$$-49x-35y=-35,-49x+84y=84$$

Me tango -49x+84y=84 mai i -49x-35y=-35 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-49x+49x-35y-84y=-35-84$$

Tāpiri -49x ki te 49x. Ka whakakore atu ngā kupu -49x me 49x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-35y-84y=-35-84$$

Tāpiri -35y ki te -84y.

$$-119y=-35-84$$

Tāpiri -35 ki te -84.

$$-119y=-119$$

Whakawehea ngā taha e rua ki te -119.

$$y=1$$

Whakaurua te 1 mō y ki -7x+12y=12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x+12=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-7x=0$$

Whakawehea ngā taha e rua ki te -7.

$$x=0$$

Kua oti te pūnaha te whakatau.

$$x=0,y=1$$