Whakaoti i te {l}{2y+7x/10=18}{7x+18y=30}

$$\left. \begin{array} { l } { 2 y + \frac { 7 x } { 10 } = 18 } \\ { 7 x + 18 y = 30 } \end{array} \right.$$

Whakaoti mō y, x

$$x = -\frac{1320}{7} = -188\frac{4}{7} \approx -188.571428571$$

$$y=75$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 10.

$$20y+7x=180$$

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.

$$20y+7x=180,18y+7x=30$$

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

$$20y+7x=180$$

Me tango 7x mai i ngā taha e rua o te whārite.

$$20y=-7x+180$$

Whakawehea ngā taha e rua ki te 20.

$$y=\frac{1}{20}\left(-7x+180\right)$$

Whakareatia \frac{1}{20} ki te -7x+180.

$$y=-\frac{7}{20}x+9$$

Whakakapia te -\frac{7x}{20}+9 mō te y ki tērā atu whārite, 18y+7x=30.

$$18\left(-\frac{7}{20}x+9\right)+7x=30$$

Whakareatia 18 ki te -\frac{7x}{20}+9.

$$-\frac{63}{10}x+162+7x=30$$

Tāpiri -\frac{63x}{10} ki te 7x.

$$\frac{7}{10}x+162=30$$

Me tango 162 mai i ngā taha e rua o te whārite.

$$\frac{7}{10}x=-132$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{10}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$x=-\frac{1320}{7}$$

Whakaurua te -\frac{1320}{7} mō x ki y=-\frac{7}{20}x+9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=-\frac{7}{20}\left(-\frac{1320}{7}\right)+9$$

Whakareatia -\frac{7}{20} ki te -\frac{1320}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$y=66+9$$

Tāpiri 9 ki te 66.

$$y=75$$

Kua oti te pūnaha te whakatau.

$$y=75,x=-\frac{1320}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 10.

$$20y+7x=180$$

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

$$20y+7x=180,18y+7x=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}20&7\\18&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}180\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}20&7\\18&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}20&7\\18&7\end{matrix}\right))\left(\begin{matrix}20&7\\18&7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&7\\18&7\end{matrix}\right))\left(\begin{matrix}180\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&7\\18&7\end{matrix}\right))\left(\begin{matrix}180\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}20&7\\18&7\end{matrix}\right))\left(\begin{matrix}180\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{7}{20\times 7-7\times 18}&-\frac{7}{20\times 7-7\times 18}\\-\frac{18}{20\times 7-7\times 18}&\frac{20}{20\times 7-7\times 18}\end{matrix}\right)\left(\begin{matrix}180\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\-\frac{9}{7}&\frac{10}{7}\end{matrix}\right)\left(\begin{matrix}180\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 180-\frac{1}{2}\times 30\\-\frac{9}{7}\times 180+\frac{10}{7}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}75\\-\frac{1320}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=75,x=-\frac{1320}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 10.

$$20y+7x=180$$

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.

$$20y+7x=180,18y+7x=30$$

Me tango 18y+7x=30 mai i 20y+7x=180 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20y-18y+7x-7x=180-30$$

Tāpiri 7x ki te -7x. Ka whakakore atu ngā kupu 7x me -7x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$20y-18y=180-30$$

Tāpiri 20y ki te -18y.

$$2y=180-30$$

Tāpiri 180 ki te -30.

$$2y=150$$

Whakawehea ngā taha e rua ki te 2.

$$y=75$$

Whakaurua te 75 mō y ki 18y+7x=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$18\times 75+7x=30$$

Whakareatia 18 ki te 75.

$$1350+7x=30$$

Me tango 1350 mai i ngā taha e rua o te whārite.

$$7x=-1320$$

Whakawehea ngā taha e rua ki te 7.

$$x=-\frac{1320}{7}$$

Kua oti te pūnaha te whakatau.

$$y=75,x=-\frac{1320}{7}$$