Whakaoti i te {l}{x-3.5y=2}{x-2y=16}

$$\left. \begin{array} { l } { x - 3.5 y = 2 } \\ { x - 2 y = 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{104}{3} = 34\frac{2}{3} \approx 34.666666667$$

$$y = \frac{28}{3} = 9\frac{1}{3} \approx 9.333333333$$

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.

$$x-3.5y=2,x-2y=16$$

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

$$x-3.5y=2$$

Me tāpiri \frac{7y}{2} ki ngā taha e rua o te whārite.

$$x=3.5y+2$$

Whakakapia te \frac{7y}{2}+2 mō te x ki tērā atu whārite, x-2y=16.

$$3.5y+2-2y=16$$

Tāpiri \frac{7y}{2} ki te -2y.

$$1.5y+2=16$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$1.5y=14$$

Whakawehea ngā taha e rua o te whārite ki te 1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{28}{3}$$

Whakaurua te \frac{28}{3} mō y ki x=3.5y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=3.5\times \frac{28}{3}+2$$

Whakareatia 3.5 ki te \frac{28}{3} 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.

$$x=\frac{98}{3}+2$$

Tāpiri 2 ki te \frac{98}{3}.

$$x=\frac{104}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{104}{3},y=\frac{28}{3}$$

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.

$$x-3.5y=2,x-2y=16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right))\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3.5\\1&-2\end{matrix}\right))\left(\begin{matrix}2\\16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{-2-\left(-3.5\right)}&-\frac{-3.5}{-2-\left(-3.5\right)}\\-\frac{1}{-2-\left(-3.5\right)}&\frac{1}{-2-\left(-3.5\right)}\end{matrix}\right)\left(\begin{matrix}2\\16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3}&\frac{7}{3}\\-\frac{2}{3}&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}2\\16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{3}\times 2+\frac{7}{3}\times 16\\-\frac{2}{3}\times 2+\frac{2}{3}\times 16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{104}{3}\\\frac{28}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{104}{3},y=\frac{28}{3}$$

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.

$$x-3.5y=2,x-2y=16$$

Me tango x-2y=16 mai i x-3.5y=2 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x-3.5y+2y=2-16$$

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

$$-3.5y+2y=2-16$$

Tāpiri -\frac{7y}{2} ki te 2y.

$$-1.5y=2-16$$

Tāpiri 2 ki te -16.

$$-1.5y=-14$$

Whakawehea ngā taha e rua o te whārite ki te -1.5, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{28}{3}$$

Whakaurua te \frac{28}{3} mō y ki x-2y=16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-2\times \frac{28}{3}=16$$

Whakareatia -2 ki te \frac{28}{3}.

$$x-\frac{56}{3}=16$$

Me tāpiri \frac{56}{3} ki ngā taha e rua o te whārite.

$$x=\frac{104}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{104}{3},y=\frac{28}{3}$$