Whakaoti i te {l}{4y-5x=-70}{x=-4y-34}

$$\left. \begin{array} { l } { 4 y - 5 x = - 70 } \\ { x = - 4 y - 34 } \end{array} \right.$$

Whakaoti mō y, x

$$x=6$$

$$y=-10$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 4y ki ngā taha e rua.

$$x+4y=-34$$

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.

$$4y-5x=-70,4y+x=-34$$

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

$$4y-5x=-70$$

Me tāpiri 5x ki ngā taha e rua o te whārite.

$$4y=5x-70$$

Whakawehea ngā taha e rua ki te 4.

$$y=\frac{1}{4}\left(5x-70\right)$$

Whakareatia \frac{1}{4} ki te -70+5x.

$$y=\frac{5}{4}x-\frac{35}{2}$$

Whakakapia te -\frac{35}{2}+\frac{5x}{4} mō te y ki tērā atu whārite, 4y+x=-34.

$$4\left(\frac{5}{4}x-\frac{35}{2}\right)+x=-34$$

Whakareatia 4 ki te -\frac{35}{2}+\frac{5x}{4}.

$$5x-70+x=-34$$

Tāpiri 5x ki te x.

$$6x-70=-34$$

Me tāpiri 70 ki ngā taha e rua o te whārite.

$$6x=36$$

Whakawehea ngā taha e rua ki te 6.

$$x=6$$

Whakaurua te 6 mō x ki y=\frac{5}{4}x-\frac{35}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=\frac{5}{4}\times 6-\frac{35}{2}$$

Whakareatia \frac{5}{4} ki te 6.

$$y=\frac{15-35}{2}$$

Tāpiri -\frac{35}{2} ki te \frac{15}{2} 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.

$$y=-10$$

Kua oti te pūnaha te whakatau.

$$y=-10,x=6$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 4y ki ngā taha e rua.

$$x+4y=-34$$

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

$$4y-5x=-70,4y+x=-34$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-5\\4&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-5\\4&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-5\\4&1\end{matrix}\right))\left(\begin{matrix}4&-5\\4&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\4&1\end{matrix}\right))\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\4&1\end{matrix}\right))\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\4&1\end{matrix}\right))\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-5\times 4\right)}&-\frac{-5}{4-\left(-5\times 4\right)}\\-\frac{4}{4-\left(-5\times 4\right)}&\frac{4}{4-\left(-5\times 4\right)}\end{matrix}\right)\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}&\frac{5}{24}\\-\frac{1}{6}&\frac{1}{6}\end{matrix}\right)\left(\begin{matrix}-70\\-34\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{1}{24}\left(-70\right)+\frac{5}{24}\left(-34\right)\\-\frac{1}{6}\left(-70\right)+\frac{1}{6}\left(-34\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-10\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-10,x=6$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 4y ki ngā taha e rua.

$$x+4y=-34$$

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.

$$4y-5x=-70,4y+x=-34$$

Me tango 4y+x=-34 mai i 4y-5x=-70 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$4y-4y-5x-x=-70+34$$

Tāpiri 4y ki te -4y. Ka whakakore atu ngā kupu 4y me -4y, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5x-x=-70+34$$

Tāpiri -5x ki te -x.

$$-6x=-70+34$$

Tāpiri -70 ki te 34.

$$-6x=-36$$

Whakawehea ngā taha e rua ki te -6.

$$x=6$$

Whakaurua te 6 mō x ki 4y+x=-34. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$4y+6=-34$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$4y=-40$$

Whakawehea ngā taha e rua ki te 4.

$$y=-10$$

Kua oti te pūnaha te whakatau.

$$y=-10,x=6$$