Whakaoti i te {l}{7x+2y=24}{8x+2y=30}

$$\left\{ \begin{array} { l } { 7 x + 2 y = 24 } \\ { 8 x + 2 y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6$$

$$y=-9$$

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.

$$7x+2y=24,8x+2y=30$$

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

$$7x+2y=24$$

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

$$7x=-2y+24$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-2y+24\right)$$

Whakareatia \frac{1}{7} ki te -2y+24.

$$x=-\frac{2}{7}y+\frac{24}{7}$$

Whakakapia te \frac{-2y+24}{7} mō te x ki tērā atu whārite, 8x+2y=30.

$$8\left(-\frac{2}{7}y+\frac{24}{7}\right)+2y=30$$

Whakareatia 8 ki te \frac{-2y+24}{7}.

$$-\frac{16}{7}y+\frac{192}{7}+2y=30$$

Tāpiri -\frac{16y}{7} ki te 2y.

$$-\frac{2}{7}y+\frac{192}{7}=30$$

Me tango \frac{192}{7} mai i ngā taha e rua o te whārite.

$$-\frac{2}{7}y=\frac{18}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{2}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-9$$

Whakaurua te -9 mō y ki x=-\frac{2}{7}y+\frac{24}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{7}\left(-9\right)+\frac{24}{7}$$

Whakareatia -\frac{2}{7} ki te -9.

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

Tāpiri \frac{24}{7} ki te \frac{18}{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.

$$x=6,y=-9$$

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.

$$7x+2y=24,8x+2y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&2\\8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&2\\8&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&2\\8&2\end{matrix}\right))\left(\begin{matrix}7&2\\8&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\8&2\end{matrix}\right))\left(\begin{matrix}24\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\8&2\end{matrix}\right))\left(\begin{matrix}24\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&2\\8&2\end{matrix}\right))\left(\begin{matrix}24\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7\times 2-2\times 8}&-\frac{2}{7\times 2-2\times 8}\\-\frac{8}{7\times 2-2\times 8}&\frac{7}{7\times 2-2\times 8}\end{matrix}\right)\left(\begin{matrix}24\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1&1\\4&-\frac{7}{2}\end{matrix}\right)\left(\begin{matrix}24\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24+30\\4\times 24-\frac{7}{2}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=6,y=-9$$

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.

$$7x+2y=24,8x+2y=30$$

Me tango 8x+2y=30 mai i 7x+2y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-8x+2y-2y=24-30$$

Tāpiri 2y ki te -2y. Ka whakakore atu ngā kupu 2y me -2y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7x-8x=24-30$$

Tāpiri 7x ki te -8x.

$$-x=24-30$$

Tāpiri 24 ki te -30.

$$-x=-6$$

Whakawehea ngā taha e rua ki te -1.

$$x=6$$

Whakaurua te 6 mō x ki 8x+2y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$8\times 6+2y=30$$

Whakareatia 8 ki te 6.

$$48+2y=30$$

Me tango 48 mai i ngā taha e rua o te whārite.

$$2y=-18$$

Whakawehea ngā taha e rua ki te 2.

$$y=-9$$

Kua oti te pūnaha te whakatau.

$$x=6,y=-9$$