Whakaoti i te {l}{4x-2y=8}{5x+3y=-1}

$$\left\{ \begin{array} { l } { 4 x - 2 y = 8 } \\ { 5 x + 3 y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=-2$$

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.

$$4x-2y=8,5x+3y=-1$$

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

$$4x-2y=8$$

Me tāpiri 2y ki ngā taha e rua o te whārite.

$$4x=2y+8$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y+8\right)$$

Whakareatia \frac{1}{4} ki te 8+2y.

$$x=\frac{1}{2}y+2$$

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

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

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

$$\frac{5}{2}y+10+3y=-1$$

Tāpiri \frac{5y}{2} ki te 3y.

$$\frac{11}{2}y+10=-1$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$\frac{11}{2}y=-11$$

Whakawehea ngā taha e rua o te whārite ki te \frac{11}{2}, 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{1}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

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

Whakareatia \frac{1}{2} ki te -2.

$$x=-1+2$$

Tāpiri 2 ki te -1.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$

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.

$$4x-2y=8,5x+3y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-2\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\5&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-2\\5&3\end{matrix}\right))\left(\begin{matrix}4&-2\\5&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\5&3\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\5&3\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\5&3\end{matrix}\right))\left(\begin{matrix}8\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-\left(-2\times 5\right)}&-\frac{-2}{4\times 3-\left(-2\times 5\right)}\\-\frac{5}{4\times 3-\left(-2\times 5\right)}&\frac{4}{4\times 3-\left(-2\times 5\right)}\end{matrix}\right)\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}&\frac{1}{11}\\-\frac{5}{22}&\frac{2}{11}\end{matrix}\right)\left(\begin{matrix}8\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{22}\times 8+\frac{1}{11}\left(-1\right)\\-\frac{5}{22}\times 8+\frac{2}{11}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=1,y=-2$$

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.

$$4x-2y=8,5x+3y=-1$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\left(-2\right)y=5\times 8,4\times 5x+4\times 3y=4\left(-1\right)$$

Whakarūnātia.

$$20x-10y=40,20x+12y=-4$$

Me tango 20x+12y=-4 mai i 20x-10y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x-10y-12y=40+4$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-10y-12y=40+4$$

Tāpiri -10y ki te -12y.

$$-22y=40+4$$

Tāpiri 40 ki te 4.

$$-22y=44$$

Whakawehea ngā taha e rua ki te -22.

$$y=-2$$

Whakaurua te -2 mō y ki 5x+3y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+3\left(-2\right)=-1$$

Whakareatia 3 ki te -2.

$$5x-6=-1$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$5x=5$$

Whakawehea ngā taha e rua ki te 5.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=-2$$