Whakaoti i te {l}{3x+5y=4}{x-3y=6}

$$\left.\begin{array} { l } { 3 x + 5 y = 4 } \\ { x - 3 y = 6 } \end{array} \right\}$$

Whakaoti mō x, y

$$x=3$$

$$y=-1$$

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.

$$3x+5y=4,x-3y=6$$

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

$$3x+5y=4$$

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

$$3x=-5y+4$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{1}{3}\left(-5y+4\right)$$

Whakareatia \frac{1}{3} ki te -5y+4.

$$x=-\frac{5}{3}y+\frac{4}{3}$$

Whakakapia te \frac{-5y+4}{3} mō te x ki tērā atu whārite, x-3y=6.

$$-\frac{5}{3}y+\frac{4}{3}-3y=6$$

Tāpiri -\frac{5y}{3} ki te -3y.

$$-\frac{14}{3}y+\frac{4}{3}=6$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$-\frac{14}{3}y=\frac{14}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{14}{3}, 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{5}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{3}\left(-1\right)+\frac{4}{3}$$

Whakareatia -\frac{5}{3} ki te -1.

$$x=\frac{5+4}{3}$$

Tāpiri \frac{4}{3} ki te \frac{5}{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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-1$$

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.

$$3x+5y=4,x-3y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&5\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&5\\1&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&5\\1&-3\end{matrix}\right))\left(\begin{matrix}3&5\\1&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-3\end{matrix}\right))\left(\begin{matrix}4\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-3\end{matrix}\right))\left(\begin{matrix}4\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&5\\1&-3\end{matrix}\right))\left(\begin{matrix}4\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{3\left(-3\right)-5}&-\frac{5}{3\left(-3\right)-5}\\-\frac{1}{3\left(-3\right)-5}&\frac{3}{3\left(-3\right)-5}\end{matrix}\right)\left(\begin{matrix}4\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}&\frac{5}{14}\\\frac{1}{14}&-\frac{3}{14}\end{matrix}\right)\left(\begin{matrix}4\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{14}\times 4+\frac{5}{14}\times 6\\\frac{1}{14}\times 4-\frac{3}{14}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-1$$

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.

$$3x+5y=4,x-3y=6$$

Kia ōrite ai a 3x me x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$3x+5y=4,3x+3\left(-3\right)y=3\times 6$$

Whakarūnātia.

$$3x+5y=4,3x-9y=18$$

Me tango 3x-9y=18 mai i 3x+5y=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+5y+9y=4-18$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y+9y=4-18$$

Tāpiri 5y ki te 9y.

$$14y=4-18$$

Tāpiri 4 ki te -18.

$$14y=-14$$

Whakawehea ngā taha e rua ki te 14.

$$y=-1$$

Whakaurua te -1 mō y ki x-3y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-3\left(-1\right)=6$$

Whakareatia -3 ki te -1.

$$x+3=6$$

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

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-1$$