Whakaoti i te {l}{1.5x-3.5y=-5}{-1.2x+2.5y=1}

$$\left\{ \begin{array} { l } { 1.5 x - 3.5 y = - 5 } \\ { - 1.2 x + 2.5 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=20$$

$$y=10$$

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.

$$1.5x-3.5y=-5,-1.2x+2.5y=1$$

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

$$1.5x-3.5y=-5$$

Me tāpiri \frac{7y}{2} ki ngā taha e rua o te whārite.

$$1.5x=3.5y-5$$

Whakawehea ngā taha e rua o te whārite ki te 1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=\frac{2}{3}\left(3.5y-5\right)$$

Whakareatia \frac{2}{3} ki te \frac{7y}{2}-5.

$$x=\frac{7}{3}y-\frac{10}{3}$$

Whakakapia te \frac{7y-10}{3} mō te x ki tērā atu whārite, -1.2x+2.5y=1.

$$-1.2\left(\frac{7}{3}y-\frac{10}{3}\right)+2.5y=1$$

Whakareatia -1.2 ki te \frac{7y-10}{3}.

$$-2.8y+4+2.5y=1$$

Tāpiri -\frac{14y}{5} ki te \frac{5y}{2}.

$$-0.3y+4=1$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$-0.3y=-3$$

Whakawehea ngā taha e rua o te whārite ki te -0.3, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki x=\frac{7}{3}y-\frac{10}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{7}{3}\times 10-\frac{10}{3}$$

Whakareatia \frac{7}{3} ki te 10.

$$x=\frac{70-10}{3}$$

Tāpiri -\frac{10}{3} ki te \frac{70}{3} 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.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=10$$

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.

$$1.5x-3.5y=-5,-1.2x+2.5y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right))\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1.5&-3.5\\-1.2&2.5\end{matrix}\right))\left(\begin{matrix}-5\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2.5}{1.5\times 2.5-\left(-3.5\left(-1.2\right)\right)}&-\frac{-3.5}{1.5\times 2.5-\left(-3.5\left(-1.2\right)\right)}\\-\frac{-1.2}{1.5\times 2.5-\left(-3.5\left(-1.2\right)\right)}&\frac{1.5}{1.5\times 2.5-\left(-3.5\left(-1.2\right)\right)}\end{matrix}\right)\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{50}{9}&-\frac{70}{9}\\-\frac{8}{3}&-\frac{10}{3}\end{matrix}\right)\left(\begin{matrix}-5\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{50}{9}\left(-5\right)-\frac{70}{9}\\-\frac{8}{3}\left(-5\right)-\frac{10}{3}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\10\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=20,y=10$$

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.

$$1.5x-3.5y=-5,-1.2x+2.5y=1$$

Kia ōrite ai a \frac{3x}{2} me -\frac{6x}{5}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1.2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.5.

$$-1.2\times 1.5x-1.2\left(-3.5\right)y=-1.2\left(-5\right),1.5\left(-1.2\right)x+1.5\times 2.5y=1.5$$

Whakarūnātia.

$$-1.8x+4.2y=6,-1.8x+3.75y=1.5$$

Me tango -1.8x+3.75y=1.5 mai i -1.8x+4.2y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-1.8x+1.8x+4.2y-3.75y=6-1.5$$

Tāpiri -\frac{9x}{5} ki te \frac{9x}{5}. Ka whakakore atu ngā kupu -\frac{9x}{5} me \frac{9x}{5}, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$4.2y-3.75y=6-1.5$$

Tāpiri \frac{21y}{5} ki te -\frac{15y}{4}.

$$0.45y=6-1.5$$

Tāpiri 6 ki te -1.5.

$$0.45y=4.5$$

Whakawehea ngā taha e rua o te whārite ki te 0.45, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=10$$

Whakaurua te 10 mō y ki -1.2x+2.5y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-1.2x+2.5\times 10=1$$

Whakareatia 2.5 ki te 10.

$$-1.2x+25=1$$

Me tango 25 mai i ngā taha e rua o te whārite.

$$-1.2x=-24$$

Whakawehea ngā taha e rua o te whārite ki te -1.2, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=20$$

Kua oti te pūnaha te whakatau.

$$x=20,y=10$$