Whakaoti i te {l}{5x+9y=40}{3x+7y=3}

$$\left\{ \begin{array} { l } { 5 x + 9 y = 40 } \\ { 3 x + 7 y = 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{253}{8} = 31\frac{5}{8} = 31.625$$

$$y = -\frac{105}{8} = -13\frac{1}{8} = -13.125$$

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.

$$5x+9y=40,3x+7y=3$$

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

$$5x+9y=40$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$5x=-9y+40$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-9y+40\right)$$

Whakareatia \frac{1}{5} ki te -9y+40.

$$x=-\frac{9}{5}y+8$$

Whakakapia te -\frac{9y}{5}+8 mō te x ki tērā atu whārite, 3x+7y=3.

$$3\left(-\frac{9}{5}y+8\right)+7y=3$$

Whakareatia 3 ki te -\frac{9y}{5}+8.

$$-\frac{27}{5}y+24+7y=3$$

Tāpiri -\frac{27y}{5} ki te 7y.

$$\frac{8}{5}y+24=3$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$\frac{8}{5}y=-21$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{105}{8}$$

Whakaurua te -\frac{105}{8} mō y ki x=-\frac{9}{5}y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{9}{5}\left(-\frac{105}{8}\right)+8$$

Whakareatia -\frac{9}{5} ki te -\frac{105}{8} 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.

$$x=\frac{189}{8}+8$$

Tāpiri 8 ki te \frac{189}{8}.

$$x=\frac{253}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{253}{8},y=-\frac{105}{8}$$

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.

$$5x+9y=40,3x+7y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&9\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&9\\3&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&9\\3&7\end{matrix}\right))\left(\begin{matrix}5&9\\3&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&9\\3&7\end{matrix}\right))\left(\begin{matrix}40\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&9\\3&7\end{matrix}\right))\left(\begin{matrix}40\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&9\\3&7\end{matrix}\right))\left(\begin{matrix}40\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{5\times 7-9\times 3}&-\frac{9}{5\times 7-9\times 3}\\-\frac{3}{5\times 7-9\times 3}&\frac{5}{5\times 7-9\times 3}\end{matrix}\right)\left(\begin{matrix}40\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{8}&-\frac{9}{8}\\-\frac{3}{8}&\frac{5}{8}\end{matrix}\right)\left(\begin{matrix}40\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{8}\times 40-\frac{9}{8}\times 3\\-\frac{3}{8}\times 40+\frac{5}{8}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{253}{8}\\-\frac{105}{8}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{253}{8},y=-\frac{105}{8}$$

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.

$$5x+9y=40,3x+7y=3$$

Kia ōrite ai a 5x 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 5.

$$3\times 5x+3\times 9y=3\times 40,5\times 3x+5\times 7y=5\times 3$$

Whakarūnātia.

$$15x+27y=120,15x+35y=15$$

Me tango 15x+35y=15 mai i 15x+27y=120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$15x-15x+27y-35y=120-15$$

Tāpiri 15x ki te -15x. Ka whakakore atu ngā kupu 15x me -15x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$27y-35y=120-15$$

Tāpiri 27y ki te -35y.

$$-8y=120-15$$

Tāpiri 120 ki te -15.

$$-8y=105$$

Whakawehea ngā taha e rua ki te -8.

$$y=-\frac{105}{8}$$

Whakaurua te -\frac{105}{8} mō y ki 3x+7y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+7\left(-\frac{105}{8}\right)=3$$

Whakareatia 7 ki te -\frac{105}{8}.

$$3x-\frac{735}{8}=3$$

Me tāpiri \frac{735}{8} ki ngā taha e rua o te whārite.

$$3x=\frac{759}{8}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{253}{8}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{253}{8},y=-\frac{105}{8}$$