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

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

Whakaoti mō x, y

$$x=4$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$y-x=-2$$

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.

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

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

$$5x+2y=24$$

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

$$5x=-2y+24$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-2y+24\right)$$

Whakareatia \frac{1}{5} ki te -2y+24.

$$x=-\frac{2}{5}y+\frac{24}{5}$$

Whakakapia te \frac{-2y+24}{5} mō te x ki tērā atu whārite, -x+y=-2.

$$-\left(-\frac{2}{5}y+\frac{24}{5}\right)+y=-2$$

Whakareatia -1 ki te \frac{-2y+24}{5}.

$$\frac{2}{5}y-\frac{24}{5}+y=-2$$

Tāpiri \frac{2y}{5} ki te y.

$$\frac{7}{5}y-\frac{24}{5}=-2$$

Me tāpiri \frac{24}{5} ki ngā taha e rua o te whārite.

$$\frac{7}{5}y=\frac{14}{5}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=-\frac{2}{5}y+\frac{24}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{2}{5}\times 2+\frac{24}{5}$$

Whakareatia -\frac{2}{5} ki te 2.

$$x=\frac{-4+24}{5}$$

Tāpiri \frac{24}{5} ki te -\frac{4}{5} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$y-x=-2$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{2}{7}\\\frac{1}{7}&\frac{5}{7}\end{matrix}\right)\left(\begin{matrix}24\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 24-\frac{2}{7}\left(-2\right)\\\frac{1}{7}\times 24+\frac{5}{7}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$y-x=-2$$

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.

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

Kia ōrite ai a 5x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 5.

$$-5x-2y=-24,5\left(-1\right)x+5y=5\left(-2\right)$$

Whakarūnātia.

$$-5x-2y=-24,-5x+5y=-10$$

Me tango -5x+5y=-10 mai i -5x-2y=-24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-5x+5x-2y-5y=-24+10$$

Tāpiri -5x ki te 5x. Ka whakakore atu ngā kupu -5x me 5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-2y-5y=-24+10$$

Tāpiri -2y ki te -5y.

$$-7y=-24+10$$

Tāpiri -24 ki te 10.

$$-7y=-14$$

Whakawehea ngā taha e rua ki te -7.

$$y=2$$

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

$$-x+2=-2$$

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

$$-x=-4$$

Whakawehea ngā taha e rua ki te -1.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$