Whakaoti i te {l}{y-5=3x}{2y=7x+9}

$$\left. \begin{array} { l } { y - 5 = 3 x } \\ { 2 y = 7 x + 9 } \end{array} \right.$$

Whakaoti mō y, x

$$x=1$$

$$y=8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-5-3x=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-3x=5$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2y-7x=9$$

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.

$$y-3x=5,2y-7x=9$$

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

$$y-3x=5$$

Me tāpiri 3x ki ngā taha e rua o te whārite.

$$y=3x+5$$

Whakakapia te 3x+5 mō te y ki tērā atu whārite, 2y-7x=9.

$$2\left(3x+5\right)-7x=9$$

Whakareatia 2 ki te 3x+5.

$$6x+10-7x=9$$

Tāpiri 6x ki te -7x.

$$-x+10=9$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$-x=-1$$

Whakawehea ngā taha e rua ki te -1.

$$x=1$$

Whakaurua te 1 mō x ki y=3x+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=3+5$$

Tāpiri 5 ki te 3.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-5-3x=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-3x=5$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2y-7x=9$$

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

$$y-3x=5,2y-7x=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}5\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\2&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right))\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\2&-7\end{matrix}\right))\left(\begin{matrix}5\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{-7-\left(-3\times 2\right)}&-\frac{-3}{-7-\left(-3\times 2\right)}\\-\frac{2}{-7-\left(-3\times 2\right)}&\frac{1}{-7-\left(-3\times 2\right)}\end{matrix}\right)\left(\begin{matrix}5\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7&-3\\2&-1\end{matrix}\right)\left(\begin{matrix}5\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}7\times 5-3\times 9\\2\times 5-9\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=8,x=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 3x mai i ngā taha e rua.

$$y-5-3x=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$y-3x=5$$

Whakaarohia te whārite tuarua. Tangohia te 7x mai i ngā taha e rua.

$$2y-7x=9$$

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.

$$y-3x=5,2y-7x=9$$

Kia ōrite ai a y me 2y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 2 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$2y+2\left(-3\right)x=2\times 5,2y-7x=9$$

Whakarūnātia.

$$2y-6x=10,2y-7x=9$$

Me tango 2y-7x=9 mai i 2y-6x=10 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$2y-2y-6x+7x=10-9$$

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.

$$-6x+7x=10-9$$

Tāpiri -6x ki te 7x.

$$x=10-9$$

Tāpiri 10 ki te -9.

$$x=1$$

Whakaurua te 1 mō x ki 2y-7x=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2y-7=9$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2y=16$$

Whakawehea ngā taha e rua ki te 2.

$$y=8$$

Kua oti te pūnaha te whakatau.

$$y=8,x=1$$