Whakaoti i te {l}{2x-y=5}{4x+6y=24}

$$\left\{ \begin{array} { l } { 2 x - y = 5 } \\ { 4 x + 6 y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{27}{8} = 3\frac{3}{8} = 3.375$$

$$y = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

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.

$$2x-y=5,4x+6y=24$$

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

$$2x-y=5$$

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

$$2x=y+5$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{5+y}{2} mō te x ki tērā atu whārite, 4x+6y=24.

$$4\left(\frac{1}{2}y+\frac{5}{2}\right)+6y=24$$

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

$$2y+10+6y=24$$

Tāpiri 2y ki te 6y.

$$8y+10=24$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$8y=14$$

Whakawehea ngā taha e rua ki te 8.

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

Whakaurua te \frac{7}{4} mō y ki x=\frac{1}{2}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\times \frac{7}{4}+\frac{5}{2}$$

Whakareatia \frac{1}{2} ki te \frac{7}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{7}{8}+\frac{5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{7}{8} 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.

$$x=\frac{27}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{8},y=\frac{7}{4}$$

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.

$$2x-y=5,4x+6y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\4&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\4&6\end{matrix}\right))\left(\begin{matrix}2&-1\\4&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\4&6\end{matrix}\right))\left(\begin{matrix}5\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\4&6\end{matrix}\right))\left(\begin{matrix}5\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\4&6\end{matrix}\right))\left(\begin{matrix}5\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-\left(-4\right)}&-\frac{-1}{2\times 6-\left(-4\right)}\\-\frac{4}{2\times 6-\left(-4\right)}&\frac{2}{2\times 6-\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}5\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}&\frac{1}{16}\\-\frac{1}{4}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}5\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{8}\times 5+\frac{1}{16}\times 24\\-\frac{1}{4}\times 5+\frac{1}{8}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{27}{8}\\\frac{7}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{27}{8},y=\frac{7}{4}$$

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.

$$2x-y=5,4x+6y=24$$

Kia ōrite ai a 2x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$4\times 2x+4\left(-1\right)y=4\times 5,2\times 4x+2\times 6y=2\times 24$$

Whakarūnātia.

$$8x-4y=20,8x+12y=48$$

Me tango 8x+12y=48 mai i 8x-4y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-4y-12y=20-48$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-4y-12y=20-48$$

Tāpiri -4y ki te -12y.

$$-16y=20-48$$

Tāpiri 20 ki te -48.

$$-16y=-28$$

Whakawehea ngā taha e rua ki te -16.

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

Whakaurua te \frac{7}{4} mō y ki 4x+6y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x+6\times \frac{7}{4}=24$$

Whakareatia 6 ki te \frac{7}{4}.

$$4x+\frac{21}{2}=24$$

Me tango \frac{21}{2} mai i ngā taha e rua o te whārite.

$$4x=\frac{27}{2}$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{27}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{27}{8},y=\frac{7}{4}$$