Whakaoti i te {l}{4x+3y=25}{x-3y/2=-14}

$$\left. \begin{array} { l } { 4 x + 3 y = 25 } \\ { x - \frac { 3 y } { 2 } = - 14 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{2}=-0.5$$

$$y=9$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-3y=-28$$

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.

$$4x+3y=25,2x-3y=-28$$

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

$$4x+3y=25$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$4x=-3y+25$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-3y+25\right)$$

Whakareatia \frac{1}{4} ki te -3y+25.

$$x=-\frac{3}{4}y+\frac{25}{4}$$

Whakakapia te \frac{-3y+25}{4} mō te x ki tērā atu whārite, 2x-3y=-28.

$$2\left(-\frac{3}{4}y+\frac{25}{4}\right)-3y=-28$$

Whakareatia 2 ki te \frac{-3y+25}{4}.

$$-\frac{3}{2}y+\frac{25}{2}-3y=-28$$

Tāpiri -\frac{3y}{2} ki te -3y.

$$-\frac{9}{2}y+\frac{25}{2}=-28$$

Me tango \frac{25}{2} mai i ngā taha e rua o te whārite.

$$-\frac{9}{2}y=-\frac{81}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{9}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=9$$

Whakaurua te 9 mō y ki x=-\frac{3}{4}y+\frac{25}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{4}\times 9+\frac{25}{4}$$

Whakareatia -\frac{3}{4} ki te 9.

$$x=\frac{-27+25}{4}$$

Tāpiri \frac{25}{4} ki te -\frac{27}{4} 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.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{2},y=9$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-3y=-28$$

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

$$4x+3y=25,2x-3y=-28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&3\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\-28\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&3\\2&-3\end{matrix}\right))\left(\begin{matrix}4&3\\2&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-3\end{matrix}\right))\left(\begin{matrix}25\\-28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-3\end{matrix}\right))\left(\begin{matrix}25\\-28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&3\\2&-3\end{matrix}\right))\left(\begin{matrix}25\\-28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4\left(-3\right)-3\times 2}&-\frac{3}{4\left(-3\right)-3\times 2}\\-\frac{2}{4\left(-3\right)-3\times 2}&\frac{4}{4\left(-3\right)-3\times 2}\end{matrix}\right)\left(\begin{matrix}25\\-28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&\frac{1}{6}\\\frac{1}{9}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}25\\-28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 25+\frac{1}{6}\left(-28\right)\\\frac{1}{9}\times 25-\frac{2}{9}\left(-28\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\\9\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{2},y=9$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$2x-3y=-28$$

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.

$$4x+3y=25,2x-3y=-28$$

Kia ōrite ai a 4x me 2x, 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 4.

$$2\times 4x+2\times 3y=2\times 25,4\times 2x+4\left(-3\right)y=4\left(-28\right)$$

Whakarūnātia.

$$8x+6y=50,8x-12y=-112$$

Me tango 8x-12y=-112 mai i 8x+6y=50 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x+6y+12y=50+112$$

Tāpiri 8x ki te -8x. Ka whakakore atu ngā kupu 8x me -8x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+12y=50+112$$

Tāpiri 6y ki te 12y.

$$18y=50+112$$

Tāpiri 50 ki te 112.

$$18y=162$$

Whakawehea ngā taha e rua ki te 18.

$$y=9$$

Whakaurua te 9 mō y ki 2x-3y=-28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x-3\times 9=-28$$

Whakareatia -3 ki te 9.

$$2x-27=-28$$

Me tāpiri 27 ki ngā taha e rua o te whārite.

$$2x=-1$$

Whakawehea ngā taha e rua ki te 2.

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

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{2},y=9$$