Whakaoti i te {l}{7x+6y=18}{x=4-y/2}

$$\left. \begin{array} { l } { 7 x + 6 y = 18 } \\ { x = 4 - \frac { y } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x+\frac{y}{2}=4$$

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x+y=8$$

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.

$$7x+6y=18,2x+y=8$$

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

$$7x+6y=18$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$7x=-6y+18$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-6y+18\right)$$

Whakareatia \frac{1}{7} ki te -6y+18.

$$x=-\frac{6}{7}y+\frac{18}{7}$$

Whakakapia te \frac{-6y+18}{7} mō te x ki tērā atu whārite, 2x+y=8.

$$2\left(-\frac{6}{7}y+\frac{18}{7}\right)+y=8$$

Whakareatia 2 ki te \frac{-6y+18}{7}.

$$-\frac{12}{7}y+\frac{36}{7}+y=8$$

Tāpiri -\frac{12y}{7} ki te y.

$$-\frac{5}{7}y+\frac{36}{7}=8$$

Me tango \frac{36}{7} mai i ngā taha e rua o te whārite.

$$-\frac{5}{7}y=\frac{20}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-4$$

Whakaurua te -4 mō y ki x=-\frac{6}{7}y+\frac{18}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{6}{7}\left(-4\right)+\frac{18}{7}$$

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

$$x=\frac{24+18}{7}$$

Tāpiri \frac{18}{7} ki te \frac{24}{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.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x+\frac{y}{2}=4$$

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x+y=8$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$7x+6y=18,2x+y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&6\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}18\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&6\\2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&6\\2&1\end{matrix}\right))\left(\begin{matrix}7&6\\2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&6\\2&1\end{matrix}\right))\left(\begin{matrix}18\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&6\\2&1\end{matrix}\right))\left(\begin{matrix}18\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&6\\2&1\end{matrix}\right))\left(\begin{matrix}18\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7-6\times 2}&-\frac{6}{7-6\times 2}\\-\frac{2}{7-6\times 2}&\frac{7}{7-6\times 2}\end{matrix}\right)\left(\begin{matrix}18\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}&\frac{6}{5}\\\frac{2}{5}&-\frac{7}{5}\end{matrix}\right)\left(\begin{matrix}18\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5}\times 18+\frac{6}{5}\times 8\\\frac{2}{5}\times 18-\frac{7}{5}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-4$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x+\frac{y}{2}=4$$

Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x+y=8$$

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.

$$7x+6y=18,2x+y=8$$

Kia ōrite ai a 7x me 2x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$2\times 7x+2\times 6y=2\times 18,7\times 2x+7y=7\times 8$$

Whakarūnātia.

$$14x+12y=36,14x+7y=56$$

Me tango 14x+7y=56 mai i 14x+12y=36 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+12y-7y=36-56$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$12y-7y=36-56$$

Tāpiri 12y ki te -7y.

$$5y=36-56$$

Tāpiri 36 ki te -56.

$$5y=-20$$

Whakawehea ngā taha e rua ki te 5.

$$y=-4$$

Whakaurua te -4 mō y ki 2x+y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-4=8$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$2x=12$$

Whakawehea ngā taha e rua ki te 2.

$$x=6$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-4$$