Whakaoti i te {l}{2(x+2)-3(y-1)=13}{3(x+2)+5(y-1)=30.9}

$$\left\{ \begin{array} { l } { 2 ( x + 2 ) - 3 ( y - 1 ) = 13 } \\ { 3 ( x + 2 ) + 5 ( y - 1 ) = 30.9 } \end{array} \right.$$

Whakaoti mō x, y

$$x=6.3$$

$$y=2.2$$

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.

$$2\left(x+2\right)-3\left(y-1\right)=13,3\left(x+2\right)+5\left(y-1\right)=30.9$$

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

$$2\left(x+2\right)-3\left(y-1\right)=13$$

Whakareatia 2 ki te x+2.

$$2x+4-3\left(y-1\right)=13$$

Whakareatia -3 ki te y-1.

$$2x+4-3y+3=13$$

Tāpiri 4 ki te 3.

$$2x-3y+7=13$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$2x-3y=6$$

Me tāpiri 3y ki ngā taha e rua o te whārite.

$$2x=3y+6$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(3y+6\right)$$

Whakareatia \frac{1}{2} ki te 6+3y.

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

Whakakapia te \frac{3y}{2}+3 mō te x ki tērā atu whārite, 3\left(x+2\right)+5\left(y-1\right)=30.9.

$$3\left(\frac{3}{2}y+3+2\right)+5\left(y-1\right)=30.9$$

Tāpiri 3 ki te 2.

$$3\left(\frac{3}{2}y+5\right)+5\left(y-1\right)=30.9$$

Whakareatia 3 ki te \frac{3y}{2}+5.

$$\frac{9}{2}y+15+5\left(y-1\right)=30.9$$

Whakareatia 5 ki te y-1.

$$\frac{9}{2}y+15+5y-5=30.9$$

Tāpiri \frac{9y}{2} ki te 5y.

$$\frac{19}{2}y+15-5=30.9$$

Tāpiri 15 ki te -5.

$$\frac{19}{2}y+10=30.9$$

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

$$\frac{19}{2}y=20.9$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{11}{5}$$

Whakaurua te \frac{11}{5} mō y ki x=\frac{3}{2}y+3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{3}{2}\times \frac{11}{5}+3$$

Whakareatia \frac{3}{2} ki te \frac{11}{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.

$$x=\frac{33}{10}+3$$

Tāpiri 3 ki te \frac{33}{10}.

$$x=\frac{63}{10}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{63}{10},y=\frac{11}{5}$$

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.

$$2\left(x+2\right)-3\left(y-1\right)=13,3\left(x+2\right)+5\left(y-1\right)=30.9$$

Whakarūnātia te whārite tuatahi ki te āhua tānga ngahuru.

$$2\left(x+2\right)-3\left(y-1\right)=13$$

Whakareatia 2 ki te x+2.

$$2x+4-3\left(y-1\right)=13$$

Whakareatia -3 ki te y-1.

$$2x+4-3y+3=13$$

Tāpiri 4 ki te 3.

$$2x-3y+7=13$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$2x-3y=6$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$3\left(x+2\right)+5\left(y-1\right)=30.9$$

Whakareatia 3 ki te x+2.

$$3x+6+5\left(y-1\right)=30.9$$

Whakareatia 5 ki te y-1.

$$3x+6+5y-5=30.9$$

Tāpiri 6 ki te -5.

$$3x+5y+1=30.9$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3x+5y=29.9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-3\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}2&-3\\3&5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\3&5\end{matrix}\right))\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{2\times 5-\left(-3\times 3\right)}&-\frac{-3}{2\times 5-\left(-3\times 3\right)}\\-\frac{3}{2\times 5-\left(-3\times 3\right)}&\frac{2}{2\times 5-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}&\frac{3}{19}\\-\frac{3}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}6\\29.9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}\times 6+\frac{3}{19}\times 29.9\\-\frac{3}{19}\times 6+\frac{2}{19}\times 29.9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{63}{10}\\\frac{11}{5}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{63}{10},y=\frac{11}{5}$$