Whakaoti i te {l}{4x+5y=0}{8x-15y=-5}

$$\left. \begin{array} { l } { 4 x + 5 y = 0 } \\ { 8 x - 15 y = - 5 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{1}{4}=-0.25$$

$$y=\frac{1}{5}=0.2$$

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.

$$4x+5y=0,8x-15y=-5$$

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

$$4x+5y=0$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$4x=-5y$$

Whakawehea ngā taha e rua ki te 4.

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

Whakareatia \frac{1}{4} ki te -5y.

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

Whakakapia te -\frac{5y}{4} mō te x ki tērā atu whārite, 8x-15y=-5.

$$8\left(-\frac{5}{4}\right)y-15y=-5$$

Whakareatia 8 ki te -\frac{5y}{4}.

$$-10y-15y=-5$$

Tāpiri -10y ki te -15y.

$$-25y=-5$$

Whakawehea ngā taha e rua ki te -25.

$$y=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō y ki x=-\frac{5}{4}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\times \frac{1}{5}$$

Whakareatia -\frac{5}{4} ki te \frac{1}{5} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{4},y=\frac{1}{5}$$

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.

$$4x+5y=0,8x-15y=-5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\8&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\8&-15\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\8&-15\end{matrix}\right))\left(\begin{matrix}4&5\\8&-15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\8&-15\end{matrix}\right))\left(\begin{matrix}0\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\8&-15\end{matrix}\right))\left(\begin{matrix}0\\-5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\8&-15\end{matrix}\right))\left(\begin{matrix}0\\-5\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{15}{4\left(-15\right)-5\times 8}&-\frac{5}{4\left(-15\right)-5\times 8}\\-\frac{8}{4\left(-15\right)-5\times 8}&\frac{4}{4\left(-15\right)-5\times 8}\end{matrix}\right)\left(\begin{matrix}0\\-5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}&\frac{1}{20}\\\frac{2}{25}&-\frac{1}{25}\end{matrix}\right)\left(\begin{matrix}0\\-5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{20}\left(-5\right)\\-\frac{1}{25}\left(-5\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{1}{4},y=\frac{1}{5}$$

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.

$$4x+5y=0,8x-15y=-5$$

Kia ōrite ai a 4x me 8x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 8 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$8\times 4x+8\times 5y=0,4\times 8x+4\left(-15\right)y=4\left(-5\right)$$

Whakarūnātia.

$$32x+40y=0,32x-60y=-20$$

Me tango 32x-60y=-20 mai i 32x+40y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$32x-32x+40y+60y=20$$

Tāpiri 32x ki te -32x. Ka whakakore atu ngā kupu 32x me -32x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40y+60y=20$$

Tāpiri 40y ki te 60y.

$$100y=20$$

Whakawehea ngā taha e rua ki te 100.

$$y=\frac{1}{5}$$

Whakaurua te \frac{1}{5} mō y ki 8x-15y=-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$8x-15\times \frac{1}{5}=-5$$

Whakareatia -15 ki te \frac{1}{5}.

$$8x-3=-5$$

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

$$8x=-2$$

Whakawehea ngā taha e rua ki te 8.

$$x=-\frac{1}{4}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{1}{4},y=\frac{1}{5}$$