Whakaoti i te {l}{4x+5y=2}{3x+4y=1}

$$\left. \begin{array} { l } { 4 x + 5 y = 2 } \\ { 3 x + 4 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=-2$$

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.

$$4x+5y=2,3x+4y=1$$

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

$$4x+5y=2$$

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

$$4x=-5y+2$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+2\right)$$

Whakareatia \frac{1}{4} ki te -5y+2.

$$x=-\frac{5}{4}y+\frac{1}{2}$$

Whakakapia te -\frac{5y}{4}+\frac{1}{2} mō te x ki tērā atu whārite, 3x+4y=1.

$$3\left(-\frac{5}{4}y+\frac{1}{2}\right)+4y=1$$

Whakareatia 3 ki te -\frac{5y}{4}+\frac{1}{2}.

$$-\frac{15}{4}y+\frac{3}{2}+4y=1$$

Tāpiri -\frac{15y}{4} ki te 4y.

$$\frac{1}{4}y+\frac{3}{2}=1$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}y=-\frac{1}{2}$$

Me whakarea ngā taha e rua ki te 4.

$$y=-2$$

Whakaurua te -2 mō y ki x=-\frac{5}{4}y+\frac{1}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\left(-2\right)+\frac{1}{2}$$

Whakareatia -\frac{5}{4} ki te -2.

$$x=\frac{5+1}{2}$$

Tāpiri \frac{1}{2} ki te \frac{5}{2} 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.

$$x=3,y=-2$$

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.

$$4x+5y=2,3x+4y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=3,y=-2$$

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.

$$4x+5y=2,3x+4y=1$$

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.

$$3\times 4x+3\times 5y=3\times 2,4\times 3x+4\times 4y=4$$

Whakarūnātia.

$$12x+15y=6,12x+16y=4$$

Me tango 12x+16y=4 mai i 12x+15y=6 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+15y-16y=6-4$$

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.

$$15y-16y=6-4$$

Tāpiri 15y ki te -16y.

$$-y=6-4$$

Tāpiri 6 ki te -4.

$$-y=2$$

Whakawehea ngā taha e rua ki te -1.

$$y=-2$$

Whakaurua te -2 mō y ki 3x+4y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+4\left(-2\right)=1$$

Whakareatia 4 ki te -2.

$$3x-8=1$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=-2$$