Whakaoti i te {l}{7x+1y=204}{x+y=24}

$$\left\{ \begin{array} { l } { 7 x + 1 y = 204 } \\ { x + y = 24 } \end{array} \right.$$

Whakaoti mō x, y

$$x=30$$

$$y=-6$$

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.

$$7x+y=204,x+y=24$$

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

$$7x+y=204$$

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

$$7x=-y+204$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-y+204\right)$$

Whakareatia \frac{1}{7} ki te -y+204.

$$x=-\frac{1}{7}y+\frac{204}{7}$$

Whakakapia te \frac{-y+204}{7} mō te x ki tērā atu whārite, x+y=24.

$$-\frac{1}{7}y+\frac{204}{7}+y=24$$

Tāpiri -\frac{y}{7} ki te y.

$$\frac{6}{7}y+\frac{204}{7}=24$$

Me tango \frac{204}{7} mai i ngā taha e rua o te whārite.

$$\frac{6}{7}y=-\frac{36}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{6}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-6$$

Whakaurua te -6 mō y ki x=-\frac{1}{7}y+\frac{204}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{7}\left(-6\right)+\frac{204}{7}$$

Whakareatia -\frac{1}{7} ki te -6.

$$x=\frac{6+204}{7}$$

Tāpiri \frac{204}{7} ki te \frac{6}{7} 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.

$$x=30$$

Kua oti te pūnaha te whakatau.

$$x=30,y=-6$$

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.

$$7x+y=204,x+y=24$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}204\\24\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&1\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&1\\1&1\end{matrix}\right))\left(\begin{matrix}7&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&1\\1&1\end{matrix}\right))\left(\begin{matrix}204\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&1\\1&1\end{matrix}\right))\left(\begin{matrix}204\\24\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&1\\1&1\end{matrix}\right))\left(\begin{matrix}204\\24\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7-1}&-\frac{1}{7-1}\\-\frac{1}{7-1}&\frac{7}{7-1}\end{matrix}\right)\left(\begin{matrix}204\\24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{1}{6}\\-\frac{1}{6}&\frac{7}{6}\end{matrix}\right)\left(\begin{matrix}204\\24\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 204-\frac{1}{6}\times 24\\-\frac{1}{6}\times 204+\frac{7}{6}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\-6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=30,y=-6$$

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.

$$7x+y=204,x+y=24$$

Me tango x+y=24 mai i 7x+y=204 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-x+y-y=204-24$$

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

$$7x-x=204-24$$

Tāpiri 7x ki te -x.

$$6x=204-24$$

Tāpiri 204 ki te -24.

$$6x=180$$

Whakawehea ngā taha e rua ki te 6.

$$x=30$$

Whakaurua te 30 mō x ki x+y=24. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$30+y=24$$

Me tango 30 mai i ngā taha e rua o te whārite.

$$y=-6$$

Kua oti te pūnaha te whakatau.

$$x=30,y=-6$$