Whakaoti i te {l}{2x+2y=10}{1/2x+3/4y=20}

$$\left\{ \begin{array} { l } { 2 x + 2 y = 10 } \\ { \frac { 1 } { 2 } x + \frac { 3 } { 4 } y = 20 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-65$$

$$y=70$$

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.

$$2x+2y=10,\frac{1}{2}x+\frac{3}{4}y=20$$

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

$$2x+2y=10$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$2x=-2y+10$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-2y+10\right)$$

Whakareatia \frac{1}{2} ki te -2y+10.

$$x=-y+5$$

Whakakapia te -y+5 mō te x ki tērā atu whārite, \frac{1}{2}x+\frac{3}{4}y=20.

$$\frac{1}{2}\left(-y+5\right)+\frac{3}{4}y=20$$

Whakareatia \frac{1}{2} ki te -y+5.

$$-\frac{1}{2}y+\frac{5}{2}+\frac{3}{4}y=20$$

Tāpiri -\frac{y}{2} ki te \frac{3y}{4}.

$$\frac{1}{4}y+\frac{5}{2}=20$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{4}y=\frac{35}{2}$$

Me whakarea ngā taha e rua ki te 4.

$$y=70$$

Whakaurua te 70 mō y ki x=-y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-70+5$$

Tāpiri 5 ki te -70.

$$x=-65$$

Kua oti te pūnaha te whakatau.

$$x=-65,y=70$$

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.

$$2x+2y=10,\frac{1}{2}x+\frac{3}{4}y=20$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\20\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&2\\\frac{1}{2}&\frac{3}{4}\end{matrix}\right))\left(\begin{matrix}10\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{\frac{3}{4}}{2\times \frac{3}{4}-2\times \frac{1}{2}}&-\frac{2}{2\times \frac{3}{4}-2\times \frac{1}{2}}\\-\frac{\frac{1}{2}}{2\times \frac{3}{4}-2\times \frac{1}{2}}&\frac{2}{2\times \frac{3}{4}-2\times \frac{1}{2}}\end{matrix}\right)\left(\begin{matrix}10\\20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}&-4\\-1&4\end{matrix}\right)\left(\begin{matrix}10\\20\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\times 10-4\times 20\\-10+4\times 20\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-65\\70\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-65,y=70$$

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.

$$2x+2y=10,\frac{1}{2}x+\frac{3}{4}y=20$$

Kia ōrite ai a 2x me \frac{x}{2}, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te \frac{1}{2} me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$\frac{1}{2}\times 2x+\frac{1}{2}\times 2y=\frac{1}{2}\times 10,2\times \frac{1}{2}x+2\times \frac{3}{4}y=2\times 20$$

Whakarūnātia.

$$x+y=5,x+\frac{3}{2}y=40$$

Me tango x+\frac{3}{2}y=40 mai i x+y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y-\frac{3}{2}y=5-40$$

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

$$y-\frac{3}{2}y=5-40$$

Tāpiri y ki te -\frac{3y}{2}.

$$-\frac{1}{2}y=5-40$$

Tāpiri 5 ki te -40.

$$-\frac{1}{2}y=-35$$

Me whakarea ngā taha e rua ki te -2.

$$y=70$$

Whakaurua te 70 mō y ki \frac{1}{2}x+\frac{3}{4}y=20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$\frac{1}{2}x+\frac{3}{4}\times 70=20$$

Whakareatia \frac{3}{4} ki te 70.

$$\frac{1}{2}x+\frac{105}{2}=20$$

Me tango \frac{105}{2} mai i ngā taha e rua o te whārite.

$$\frac{1}{2}x=-\frac{65}{2}$$

Me whakarea ngā taha e rua ki te 2.

$$x=-65$$

Kua oti te pūnaha te whakatau.

$$x=-65,y=70$$