Whakaoti i te {l}{7x+18y=43}{2(x-3)+5=y-1}

$$\left\{ \begin{array} { l } { 7 x + 18 y = 43 } \\ { 2 ( x - 3 ) + 5 = y - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6+5=y-1$$

Tāpirihia te -6 ki te 5, ka -1.

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

Tangohia te y mai i ngā taha e rua.

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

Me tāpiri te 1 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 1, ka 0.

$$2x-y=0$$

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.

$$7x+18y=43,2x-y=0$$

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

$$7x+18y=43$$

Me tango 18y mai i ngā taha e rua o te whārite.

$$7x=-18y+43$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{1}{7}\left(-18y+43\right)$$

Whakareatia \frac{1}{7} ki te -18y+43.

$$x=-\frac{18}{7}y+\frac{43}{7}$$

Whakakapia te \frac{-18y+43}{7} mō te x ki tērā atu whārite, 2x-y=0.

$$2\left(-\frac{18}{7}y+\frac{43}{7}\right)-y=0$$

Whakareatia 2 ki te \frac{-18y+43}{7}.

$$-\frac{36}{7}y+\frac{86}{7}-y=0$$

Tāpiri -\frac{36y}{7} ki te -y.

$$-\frac{43}{7}y+\frac{86}{7}=0$$

Me tango \frac{86}{7} mai i ngā taha e rua o te whārite.

$$-\frac{43}{7}y=-\frac{86}{7}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{43}{7}, 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{18}{7}y+\frac{43}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{18}{7}\times 2+\frac{43}{7}$$

Whakareatia -\frac{18}{7} ki te 2.

$$x=\frac{-36+43}{7}$$

Tāpiri \frac{43}{7} ki te -\frac{36}{7} 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.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6+5=y-1$$

Tāpirihia te -6 ki te 5, ka -1.

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

Tangohia te y mai i ngā taha e rua.

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

Me tāpiri te 1 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 1, ka 0.

$$2x-y=0$$

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

$$7x+18y=43,2x-y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}7&18\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}43\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}7&18\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}7&18\\2&-1\end{matrix}\right))\left(\begin{matrix}7&18\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&18\\2&-1\end{matrix}\right))\left(\begin{matrix}43\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&18\\2&-1\end{matrix}\right))\left(\begin{matrix}43\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}7&18\\2&-1\end{matrix}\right))\left(\begin{matrix}43\\0\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7\left(-1\right)-18\times 2}&-\frac{18}{7\left(-1\right)-18\times 2}\\-\frac{2}{7\left(-1\right)-18\times 2}&\frac{7}{7\left(-1\right)-18\times 2}\end{matrix}\right)\left(\begin{matrix}43\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{43}&\frac{18}{43}\\\frac{2}{43}&-\frac{7}{43}\end{matrix}\right)\left(\begin{matrix}43\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{43}\times 43\\\frac{2}{43}\times 43\end{matrix}\right)$$

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

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x-3.

$$2x-6+5=y-1$$

Tāpirihia te -6 ki te 5, ka -1.

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

Tangohia te y mai i ngā taha e rua.

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

Me tāpiri te 1 ki ngā taha e rua.

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

Tāpirihia te -1 ki te 1, ka 0.

$$2x-y=0$$

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.

$$7x+18y=43,2x-y=0$$

Kia ōrite ai a 7x 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 7.

$$2\times 7x+2\times 18y=2\times 43,7\times 2x+7\left(-1\right)y=0$$

Whakarūnātia.

$$14x+36y=86,14x-7y=0$$

Me tango 14x-7y=0 mai i 14x+36y=86 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$14x-14x+36y+7y=86$$

Tāpiri 14x ki te -14x. Ka whakakore atu ngā kupu 14x me -14x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$36y+7y=86$$

Tāpiri 36y ki te 7y.

$$43y=86$$

Whakawehea ngā taha e rua ki te 43.

$$y=2$$

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

$$2x-2=0$$

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

$$2x=2$$

Whakawehea ngā taha e rua ki te 2.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=2$$