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

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

Whakaoti mō x, y

$$x = \frac{60}{19} = 3\frac{3}{19} \approx 3.157894737$$

$$y = \frac{124}{19} = 6\frac{10}{19} \approx 6.526315789$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$8x-3y-y=-4+x$$

Pahekotia te -3y me -y, ka -4y.

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

Tangohia te x mai i ngā taha e rua.

$$8x-4y-x=-4$$

Pahekotia te 8x me -x, ka 7x.

$$7x-4y=-4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2x-1.

$$-10x+5+3\left(2y+2\right)=-1+3y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2y+2.

$$-10x+5+6y+6=-1+3y$$

Tāpirihia te 5 ki te 6, ka 11.

$$-10x+11+6y=-1+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$-10x+11+6y-3y=-1$$

Pahekotia te 6y me -3y, ka 3y.

$$-10x+11+3y=-1$$

Tangohia te 11 mai i ngā taha e rua.

$$-10x+3y=-1-11$$

Tangohia te 11 i te -1, ka -12.

$$-10x+3y=-12$$

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.

$$7x-4y=-4,-10x+3y=-12$$

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

$$7x-4y=-4$$

Me tāpiri 4y ki ngā taha e rua o te whārite.

$$7x=4y-4$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(4y-4\right)$$

Whakareatia \frac{1}{7} ki te -4+4y.

$$x=\frac{4}{7}y-\frac{4}{7}$$

Whakakapia te \frac{-4+4y}{7} mō te x ki tērā atu whārite, -10x+3y=-12.

$$-10\left(\frac{4}{7}y-\frac{4}{7}\right)+3y=-12$$

Whakareatia -10 ki te \frac{-4+4y}{7}.

$$-\frac{40}{7}y+\frac{40}{7}+3y=-12$$

Tāpiri -\frac{40y}{7} ki te 3y.

$$-\frac{19}{7}y+\frac{40}{7}=-12$$

Me tango \frac{40}{7} mai i ngā taha e rua o te whārite.

$$-\frac{19}{7}y=-\frac{124}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{19}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{124}{19}$$

Whakaurua te \frac{124}{19} mō y ki x=\frac{4}{7}y-\frac{4}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{4}{7}\times \frac{124}{19}-\frac{4}{7}$$

Whakareatia \frac{4}{7} ki te \frac{124}{19} 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.

$$x=\frac{496}{133}-\frac{4}{7}$$

Tāpiri -\frac{4}{7} ki te \frac{496}{133} 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.

$$x=\frac{60}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{60}{19},y=\frac{124}{19}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$8x-3y-y=-4+x$$

Pahekotia te -3y me -y, ka -4y.

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

Tangohia te x mai i ngā taha e rua.

$$8x-4y-x=-4$$

Pahekotia te 8x me -x, ka 7x.

$$7x-4y=-4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2x-1.

$$-10x+5+3\left(2y+2\right)=-1+3y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2y+2.

$$-10x+5+6y+6=-1+3y$$

Tāpirihia te 5 ki te 6, ka 11.

$$-10x+11+6y=-1+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$-10x+11+6y-3y=-1$$

Pahekotia te 6y me -3y, ka 3y.

$$-10x+11+3y=-1$$

Tangohia te 11 mai i ngā taha e rua.

$$-10x+3y=-1-11$$

Tangohia te 11 i te -1, ka -12.

$$-10x+3y=-12$$

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

$$7x-4y=-4,-10x+3y=-12$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&-4\\-10&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right))\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right))\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right))\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&-4\\-10&3\end{matrix}\right))\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{7\times 3-\left(-4\left(-10\right)\right)}&-\frac{-4}{7\times 3-\left(-4\left(-10\right)\right)}\\-\frac{-10}{7\times 3-\left(-4\left(-10\right)\right)}&\frac{7}{7\times 3-\left(-4\left(-10\right)\right)}\end{matrix}\right)\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}&-\frac{4}{19}\\-\frac{10}{19}&-\frac{7}{19}\end{matrix}\right)\left(\begin{matrix}-4\\-12\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{19}\left(-4\right)-\frac{4}{19}\left(-12\right)\\-\frac{10}{19}\left(-4\right)-\frac{7}{19}\left(-12\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{60}{19}\\\frac{124}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{60}{19},y=\frac{124}{19}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te y mai i ngā taha e rua.

$$8x-3y-y=-4+x$$

Pahekotia te -3y me -y, ka -4y.

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

Tangohia te x mai i ngā taha e rua.

$$8x-4y-x=-4$$

Pahekotia te 8x me -x, ka 7x.

$$7x-4y=-4$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -5 ki te 2x-1.

$$-10x+5+3\left(2y+2\right)=-1+3y$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 2y+2.

$$-10x+5+6y+6=-1+3y$$

Tāpirihia te 5 ki te 6, ka 11.

$$-10x+11+6y=-1+3y$$

Tangohia te 3y mai i ngā taha e rua.

$$-10x+11+6y-3y=-1$$

Pahekotia te 6y me -3y, ka 3y.

$$-10x+11+3y=-1$$

Tangohia te 11 mai i ngā taha e rua.

$$-10x+3y=-1-11$$

Tangohia te 11 i te -1, ka -12.

$$-10x+3y=-12$$

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.

$$7x-4y=-4,-10x+3y=-12$$

Kia ōrite ai a 7x me -10x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -10 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 7.

$$-10\times 7x-10\left(-4\right)y=-10\left(-4\right),7\left(-10\right)x+7\times 3y=7\left(-12\right)$$

Whakarūnātia.

$$-70x+40y=40,-70x+21y=-84$$

Me tango -70x+21y=-84 mai i -70x+40y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-70x+70x+40y-21y=40+84$$

Tāpiri -70x ki te 70x. Ka whakakore atu ngā kupu -70x me 70x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$40y-21y=40+84$$

Tāpiri 40y ki te -21y.

$$19y=40+84$$

Tāpiri 40 ki te 84.

$$19y=124$$

Whakawehea ngā taha e rua ki te 19.

$$y=\frac{124}{19}$$

Whakaurua te \frac{124}{19} mō y ki -10x+3y=-12. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-10x+3\times \frac{124}{19}=-12$$

Whakareatia 3 ki te \frac{124}{19}.

$$-10x+\frac{372}{19}=-12$$

Me tango \frac{372}{19} mai i ngā taha e rua o te whārite.

$$-10x=-\frac{600}{19}$$

Whakawehea ngā taha e rua ki te -10.

$$x=\frac{60}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{60}{19},y=\frac{124}{19}$$