Whakaoti i te {l}{-4x+5y=24}{y=2x}

$$\left. \begin{array} { l } { - 4 x + 5 y = 24 } \\ { y = 2 x } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=8$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$-4x+5y=24,-2x+y=0$$

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

$$-4x+5y=24$$

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

$$-4x=-5y+24$$

Whakawehea ngā taha e rua ki te -4.

$$x=-\frac{1}{4}\left(-5y+24\right)$$

Whakareatia -\frac{1}{4} ki te -5y+24.

$$x=\frac{5}{4}y-6$$

Whakakapia te \frac{5y}{4}-6 mō te x ki tērā atu whārite, -2x+y=0.

$$-2\left(\frac{5}{4}y-6\right)+y=0$$

Whakareatia -2 ki te \frac{5y}{4}-6.

$$-\frac{5}{2}y+12+y=0$$

Tāpiri -\frac{5y}{2} ki te y.

$$-\frac{3}{2}y+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-\frac{3}{2}y=-12$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=8$$

Whakaurua te 8 mō y ki x=\frac{5}{4}y-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4}\times 8-6$$

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

$$x=10-6$$

Tāpiri -6 ki te 10.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=8$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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

$$-4x+5y=24,-2x+y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-4&5\\-2&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right))\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right))\left(\begin{matrix}24\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right))\left(\begin{matrix}24\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-4&5\\-2&1\end{matrix}\right))\left(\begin{matrix}24\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{-4-5\left(-2\right)}&-\frac{5}{-4-5\left(-2\right)}\\-\frac{-2}{-4-5\left(-2\right)}&-\frac{4}{-4-5\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}24\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}&-\frac{5}{6}\\\frac{1}{3}&-\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}24\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{6}\times 24\\\frac{1}{3}\times 24\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\8\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=8$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=0$$

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.

$$-4x+5y=24,-2x+y=0$$

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.

$$-2\left(-4\right)x-2\times 5y=-2\times 24,-4\left(-2\right)x-4y=0$$

Whakarūnātia.

$$8x-10y=-48,8x-4y=0$$

Me tango 8x-4y=0 mai i 8x-10y=-48 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-8x-10y+4y=-48$$

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.

$$-10y+4y=-48$$

Tāpiri -10y ki te 4y.

$$-6y=-48$$

Whakawehea ngā taha e rua ki te -6.

$$y=8$$

Whakaurua te 8 mō y ki -2x+y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-2x+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-2x=-8$$

Whakawehea ngā taha e rua ki te -2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=8$$