Whakaoti i te {l}{7x=y-39}{11x=y+9}

$$\left. \begin{array} { l } { 7 x = y - 39 } \\ { 11 x = y + 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=123$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$7x-y=-39$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$11x-y=9$$

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.

$$7x-y=-39,11x-y=9$$

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

$$7x-y=-39$$

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

$$7x=y-39$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(y-39\right)$$

Whakareatia \frac{1}{7} ki te y-39.

$$x=\frac{1}{7}y-\frac{39}{7}$$

Whakakapia te \frac{-39+y}{7} mō te x ki tērā atu whārite, 11x-y=9.

$$11\left(\frac{1}{7}y-\frac{39}{7}\right)-y=9$$

Whakareatia 11 ki te \frac{-39+y}{7}.

$$\frac{11}{7}y-\frac{429}{7}-y=9$$

Tāpiri \frac{11y}{7} ki te -y.

$$\frac{4}{7}y-\frac{429}{7}=9$$

Me tāpiri \frac{429}{7} ki ngā taha e rua o te whārite.

$$\frac{4}{7}y=\frac{492}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{4}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=123$$

Whakaurua te 123 mō y ki x=\frac{1}{7}y-\frac{39}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{7}\times 123-\frac{39}{7}$$

Whakareatia \frac{1}{7} ki te 123.

$$x=\frac{123-39}{7}$$

Tāpiri -\frac{39}{7} ki te \frac{123}{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.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=123$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$7x-y=-39$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$11x-y=9$$

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

$$7x-y=-39,11x-y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-1\\11&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-1\\11&-1\end{matrix}\right))\left(\begin{matrix}-39\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7\left(-1\right)-\left(-11\right)}&-\frac{-1}{7\left(-1\right)-\left(-11\right)}\\-\frac{11}{7\left(-1\right)-\left(-11\right)}&\frac{7}{7\left(-1\right)-\left(-11\right)}\end{matrix}\right)\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\-\frac{11}{4}&\frac{7}{4}\end{matrix}\right)\left(\begin{matrix}-39\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\left(-39\right)+\frac{1}{4}\times 9\\-\frac{11}{4}\left(-39\right)+\frac{7}{4}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\123\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=123$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$7x-y=-39$$

Whakaarohia te whārite tuarua. Tangohia te y mai i ngā taha e rua.

$$11x-y=9$$

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.

$$7x-y=-39,11x-y=9$$

Me tango 11x-y=9 mai i 7x-y=-39 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$7x-11x-y+y=-39-9$$

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.

$$7x-11x=-39-9$$

Tāpiri 7x ki te -11x.

$$-4x=-39-9$$

Tāpiri -39 ki te -9.

$$-4x=-48$$

Whakawehea ngā taha e rua ki te -4.

$$x=12$$

Whakaurua te 12 mō x ki 11x-y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$11\times 12-y=9$$

Whakareatia 11 ki te 12.

$$132-y=9$$

Me tango 132 mai i ngā taha e rua o te whārite.

$$-y=-123$$

Whakawehea ngā taha e rua ki te -1.

$$y=123$$

Kua oti te pūnaha te whakatau.

$$x=12,y=123$$