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

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

Whakaoti mō x, y

$$x = -\frac{7}{5} = -1\frac{2}{5} = -1.4$$

$$y = \frac{18}{5} = 3\frac{3}{5} = 3.6$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

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

Tangohia te 4 i te -5, ka -9.

$$2x-9+3y=-1$$

Me tāpiri te 9 ki ngā taha e rua.

$$2x+3y=-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$2x+3y=8$$

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

$$y-x=5$$

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.

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

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

$$2x+3y=8$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$2x=-3y+8$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+8\right)$$

Whakareatia \frac{1}{2} ki te -3y+8.

$$x=-\frac{3}{2}y+4$$

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

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

Whakareatia -1 ki te -\frac{3y}{2}+4.

$$\frac{3}{2}y-4+y=5$$

Tāpiri \frac{3y}{2} ki te y.

$$\frac{5}{2}y-4=5$$

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

$$\frac{5}{2}y=9$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki x=-\frac{3}{2}y+4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times \frac{18}{5}+4$$

Whakareatia -\frac{3}{2} ki te \frac{18}{5} 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.

$$x=-\frac{27}{5}+4$$

Tāpiri 4 ki te -\frac{27}{5}.

$$x=-\frac{7}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7}{5},y=\frac{18}{5}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

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

Tangohia te 4 i te -5, ka -9.

$$2x-9+3y=-1$$

Me tāpiri te 9 ki ngā taha e rua.

$$2x+3y=-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$2x+3y=8$$

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

$$y-x=5$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\-1&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{5}\\\frac{18}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{7}{5},y=\frac{18}{5}$$

Ngā Upane e Whakamahi i te Tangohanga

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

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

Tangohia te 4 i te -5, ka -9.

$$2x-9+3y=-1$$

Me tāpiri te 9 ki ngā taha e rua.

$$2x+3y=-1+9$$

Tāpirihia te -1 ki te 9, ka 8.

$$2x+3y=8$$

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

$$y-x=5$$

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.

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

Kia ōrite ai a 2x 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 2.

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

Whakarūnātia.

$$-2x-3y=-8,-2x+2y=10$$

Me tango -2x+2y=10 mai i -2x-3y=-8 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-2x+2x-3y-2y=-8-10$$

Tāpiri -2x ki te 2x. Ka whakakore atu ngā kupu -2x me 2x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y-2y=-8-10$$

Tāpiri -3y ki te -2y.

$$-5y=-8-10$$

Tāpiri -8 ki te -10.

$$-5y=-18$$

Whakawehea ngā taha e rua ki te -5.

$$y=\frac{18}{5}$$

Whakaurua te \frac{18}{5} mō y ki -x+y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+\frac{18}{5}=5$$

Me tango \frac{18}{5} mai i ngā taha e rua o te whārite.

$$-x=\frac{7}{5}$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\frac{7}{5}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{7}{5},y=\frac{18}{5}$$