Whakaoti i te {l}{-x+3/4y=7}{4x-y=-16}

$$\left. \begin{array} { l } { - x + \frac { 3 } { 4 } y = 7 } \\ { 4 x - y = - 16 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$y=6$$

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.

$$-x+\frac{3}{4}y=7,4x-y=-16$$

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

$$-x+\frac{3}{4}y=7$$

Me tango \frac{3y}{4} mai i ngā taha e rua o te whārite.

$$-x=-\frac{3}{4}y+7$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-\frac{3}{4}y+7\right)$$

Whakareatia -1 ki te -\frac{3y}{4}+7.

$$x=\frac{3}{4}y-7$$

Whakakapia te \frac{3y}{4}-7 mō te x ki tērā atu whārite, 4x-y=-16.

$$4\left(\frac{3}{4}y-7\right)-y=-16$$

Whakareatia 4 ki te \frac{3y}{4}-7.

$$3y-28-y=-16$$

Tāpiri 3y ki te -y.

$$2y-28=-16$$

Me tāpiri 28 ki ngā taha e rua o te whārite.

$$2y=12$$

Whakawehea ngā taha e rua ki te 2.

$$y=6$$

Whakaurua te 6 mō y ki x=\frac{3}{4}y-7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{4}\times 6-7$$

Whakareatia \frac{3}{4} ki te 6.

$$x=\frac{9}{2}-7$$

Tāpiri -7 ki te \frac{9}{2}.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=6$$

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.

$$-x+\frac{3}{4}y=7,4x-y=-16$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\-16\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right))\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\-16\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&\frac{3}{4}\\4&-1\end{matrix}\right))\left(\begin{matrix}7\\-16\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-\left(-1\right)-\frac{3}{4}\times 4}&-\frac{\frac{3}{4}}{-\left(-1\right)-\frac{3}{4}\times 4}\\-\frac{4}{-\left(-1\right)-\frac{3}{4}\times 4}&-\frac{1}{-\left(-1\right)-\frac{3}{4}\times 4}\end{matrix}\right)\left(\begin{matrix}7\\-16\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{3}{8}\\2&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}7\\-16\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 7+\frac{3}{8}\left(-16\right)\\2\times 7+\frac{1}{2}\left(-16\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2}\\6\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{5}{2},y=6$$

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.

$$-x+\frac{3}{4}y=7,4x-y=-16$$

Kia ōrite ai a -x me 4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -1.

$$4\left(-1\right)x+4\times \frac{3}{4}y=4\times 7,-4x-\left(-y\right)=-\left(-16\right)$$

Whakarūnātia.

$$-4x+3y=28,-4x+y=16$$

Me tango -4x+y=16 mai i -4x+3y=28 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+3y-y=28-16$$

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

$$3y-y=28-16$$

Tāpiri 3y ki te -y.

$$2y=28-16$$

Tāpiri 28 ki te -16.

$$2y=12$$

Whakawehea ngā taha e rua ki te 2.

$$y=6$$

Whakaurua te 6 mō y ki 4x-y=-16. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$4x-6=-16$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$4x=-10$$

Whakawehea ngā taha e rua ki te 4.

$$x=-\frac{5}{2}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{5}{2},y=6$$