Whakaoti i te {l}{7x+8y=15}{9x+8y=1}

$$\left. \begin{array} { l } { 7 x + 8 y = 15 } \\ { 9 x + 8 y = 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-7$$

$$y=8$$

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.

$$7x+8y=15,9x+8y=1$$

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

$$7x+8y=15$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$7x=-8y+15$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-8y+15\right)$$

Whakareatia \frac{1}{7} ki te -8y+15.

$$x=-\frac{8}{7}y+\frac{15}{7}$$

Whakakapia te \frac{-8y+15}{7} mō te x ki tērā atu whārite, 9x+8y=1.

$$9\left(-\frac{8}{7}y+\frac{15}{7}\right)+8y=1$$

Whakareatia 9 ki te \frac{-8y+15}{7}.

$$-\frac{72}{7}y+\frac{135}{7}+8y=1$$

Tāpiri -\frac{72y}{7} ki te 8y.

$$-\frac{16}{7}y+\frac{135}{7}=1$$

Me tango \frac{135}{7} mai i ngā taha e rua o te whārite.

$$-\frac{16}{7}y=-\frac{128}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{16}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=8$$

Whakaurua te 8 mō y ki x=-\frac{8}{7}y+\frac{15}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{8}{7}\times 8+\frac{15}{7}$$

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

$$x=\frac{-64+15}{7}$$

Tāpiri \frac{15}{7} ki te -\frac{64}{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.

$$x=-7$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=8$$

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.

$$7x+8y=15,9x+8y=1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&8\\9&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&8\\9&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&8\\9&8\end{matrix}\right))\left(\begin{matrix}7&8\\9&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\9&8\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\9&8\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&8\\9&8\end{matrix}\right))\left(\begin{matrix}15\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{7\times 8-8\times 9}&-\frac{8}{7\times 8-8\times 9}\\-\frac{9}{7\times 8-8\times 9}&\frac{7}{7\times 8-8\times 9}\end{matrix}\right)\left(\begin{matrix}15\\1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{9}{16}&-\frac{7}{16}\end{matrix}\right)\left(\begin{matrix}15\\1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 15+\frac{1}{2}\\\frac{9}{16}\times 15-\frac{7}{16}\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-7\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-7,y=8$$

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.

$$7x+8y=15,9x+8y=1$$

Me tango 9x+8y=1 mai i 7x+8y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-9x+8y-8y=15-1$$

Tāpiri 8y ki te -8y. Ka whakakore atu ngā kupu 8y me -8y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7x-9x=15-1$$

Tāpiri 7x ki te -9x.

$$-2x=15-1$$

Tāpiri 15 ki te -1.

$$-2x=14$$

Whakawehea ngā taha e rua ki te -2.

$$x=-7$$

Whakaurua te -7 mō x ki 9x+8y=1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$9\left(-7\right)+8y=1$$

Whakareatia 9 ki te -7.

$$-63+8y=1$$

Me tāpiri 63 ki ngā taha e rua o te whārite.

$$8y=64$$

Whakawehea ngā taha e rua ki te 8.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$x=-7,y=8$$