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

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

Whakaoti mō x, y

$$x=4$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5,4.

$$10\left(x+2\right)+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+2.

$$10x+20+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$10x+20+4y-20=5x+20$$

Tangohia te 20 i te 20, ka 0.

$$10x+4y=5x+20$$

Tangohia te 5x mai i ngā taha e rua.

$$10x+4y-5x=20$$

Pahekotia te 10x me -5x, ka 5x.

$$5x+4y=20$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x+3y=x-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$3x+3y=x+8$$

Tangohia te x mai i ngā taha e rua.

$$3x+3y-x=8$$

Pahekotia te 3x me -x, ka 2x.

$$2x+3y=8$$

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.

$$5x+4y=20,2x+3y=8$$

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

$$5x+4y=20$$

Me tango 4y mai i ngā taha e rua o te whārite.

$$5x=-4y+20$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-4y+20\right)$$

Whakareatia \frac{1}{5} ki te -4y+20.

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

Whakakapia te -\frac{4y}{5}+4 mō te x ki tērā atu whārite, 2x+3y=8.

$$2\left(-\frac{4}{5}y+4\right)+3y=8$$

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

$$-\frac{8}{5}y+8+3y=8$$

Tāpiri -\frac{8y}{5} ki te 3y.

$$\frac{7}{5}y+8=8$$

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

$$\frac{7}{5}y=0$$

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.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{4}{5}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5,4.

$$10\left(x+2\right)+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+2.

$$10x+20+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$10x+20+4y-20=5x+20$$

Tangohia te 20 i te 20, ka 0.

$$10x+4y=5x+20$$

Tangohia te 5x mai i ngā taha e rua.

$$10x+4y-5x=20$$

Pahekotia te 10x me -5x, ka 5x.

$$5x+4y=20$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x+3y=x-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$3x+3y=x+8$$

Tangohia te x mai i ngā taha e rua.

$$3x+3y-x=8$$

Pahekotia te 3x me -x, ka 2x.

$$2x+3y=8$$

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

$$5x+4y=20,2x+3y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\8\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}5&4\\2&3\end{matrix}\right))\left(\begin{matrix}5&4\\2&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&4\\2&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5\times 3-4\times 2}&-\frac{4}{5\times 3-4\times 2}\\-\frac{2}{5\times 3-4\times 2}&\frac{5}{5\times 3-4\times 2}\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}&-\frac{4}{7}\\-\frac{2}{7}&\frac{5}{7}\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7}\times 20-\frac{4}{7}\times 8\\-\frac{2}{7}\times 20+\frac{5}{7}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 2,5,4.

$$10\left(x+2\right)+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+2.

$$10x+20+4\left(y-5\right)=5x+20$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te y-5.

$$10x+20+4y-20=5x+20$$

Tangohia te 20 i te 20, ka 0.

$$10x+4y=5x+20$$

Tangohia te 5x mai i ngā taha e rua.

$$10x+4y-5x=20$$

Pahekotia te 10x me -5x, ka 5x.

$$5x+4y=20$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 3.

$$3x+3y=x-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$3x+3y=x+8$$

Tangohia te x mai i ngā taha e rua.

$$3x+3y-x=8$$

Pahekotia te 3x me -x, ka 2x.

$$2x+3y=8$$

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.

$$5x+4y=20,2x+3y=8$$

Kia ōrite ai a 5x 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 5.

$$2\times 5x+2\times 4y=2\times 20,5\times 2x+5\times 3y=5\times 8$$

Whakarūnātia.

$$10x+8y=40,10x+15y=40$$

Me tango 10x+15y=40 mai i 10x+8y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x+8y-15y=40-40$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$8y-15y=40-40$$

Tāpiri 8y ki te -15y.

$$-7y=40-40$$

Tāpiri 40 ki te -40.

$$-7y=0$$

Whakawehea ngā taha e rua ki te -7.

$$y=0$$

Whakaurua te 0 mō y ki 2x+3y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$2x=8$$

Whakawehea ngā taha e rua ki te 2.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=0$$