Whakaoti i te {l}{x+y=69}{7x+y=87}

$$\left. \begin{array} { l } { x + y = 69 } \\ { 7 x + y = 87 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=66$$

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.

$$x+y=69,7x+y=87$$

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

$$x+y=69$$

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

$$x=-y+69$$

Whakakapia te -y+69 mō te x ki tērā atu whārite, 7x+y=87.

$$7\left(-y+69\right)+y=87$$

Whakareatia 7 ki te -y+69.

$$-7y+483+y=87$$

Tāpiri -7y ki te y.

$$-6y+483=87$$

Me tango 483 mai i ngā taha e rua o te whārite.

$$-6y=-396$$

Whakawehea ngā taha e rua ki te -6.

$$y=66$$

Whakaurua te 66 mō y ki x=-y+69. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-66+69$$

Tāpiri 69 ki te -66.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=66$$

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.

$$x+y=69,7x+y=87$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\7&1\end{matrix}\right))\left(\begin{matrix}1&1\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\7&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\7&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\7&1\end{matrix}\right))\left(\begin{matrix}69\\87\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-7}&-\frac{1}{1-7}\\-\frac{7}{1-7}&\frac{1}{1-7}\end{matrix}\right)\left(\begin{matrix}69\\87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{6}\\\frac{7}{6}&-\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}69\\87\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 69+\frac{1}{6}\times 87\\\frac{7}{6}\times 69-\frac{1}{6}\times 87\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\66\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=66$$

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.

$$x+y=69,7x+y=87$$

Me tango 7x+y=87 mai i x+y=69 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-7x+y-y=69-87$$

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.

$$x-7x=69-87$$

Tāpiri x ki te -7x.

$$-6x=69-87$$

Tāpiri 69 ki te -87.

$$-6x=-18$$

Whakawehea ngā taha e rua ki te -6.

$$x=3$$

Whakaurua te 3 mō x ki 7x+y=87. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$7\times 3+y=87$$

Whakareatia 7 ki te 3.

$$21+y=87$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$y=66$$

Kua oti te pūnaha te whakatau.

$$x=3,y=66$$