Whakaoti i te {l}{4x-2y-6=0}{4(8+x)+2(20y)-26=0}

$$\left\{ \begin{array} { l } { 4 x - 2 y - 6 = 0 } \\ { 4 ( 8 + x ) + 2 ( 20 y ) - 26 = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{19}{14} = 1\frac{5}{14} \approx 1.357142857$$

$$y=-\frac{2}{7}\approx -0.285714286$$

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.

$$4x-2y-6=0,4\left(x+8\right)+40y-26=0$$

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

$$4x-2y-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$4x-2y=6$$

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

$$4x=2y+6$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y+6\right)$$

Whakareatia \frac{1}{4} ki te 6+2y.

$$x=\frac{1}{2}y+\frac{3}{2}$$

Whakakapia te \frac{3+y}{2} mō te x ki tērā atu whārite, 4\left(x+8\right)+40y-26=0.

$$4\left(\frac{1}{2}y+\frac{3}{2}+8\right)+40y-26=0$$

Tāpiri \frac{3}{2} ki te 8.

$$4\left(\frac{1}{2}y+\frac{19}{2}\right)+40y-26=0$$

Whakareatia 4 ki te \frac{19+y}{2}.

$$2y+38+40y-26=0$$

Tāpiri 2y ki te 40y.

$$42y+38-26=0$$

Tāpiri 38 ki te -26.

$$42y+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$42y=-12$$

Whakawehea ngā taha e rua ki te 42.

$$y=-\frac{2}{7}$$

Whakaurua te -\frac{2}{7} mō y ki x=\frac{1}{2}y+\frac{3}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-\frac{2}{7}\right)+\frac{3}{2}$$

Whakareatia \frac{1}{2} ki te -\frac{2}{7} 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.

$$x=-\frac{1}{7}+\frac{3}{2}$$

Tāpiri \frac{3}{2} ki te -\frac{1}{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.

$$x=\frac{19}{14}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{19}{14},y=-\frac{2}{7}$$

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.

$$4x-2y-6=0,4\left(x+8\right)+40y-26=0$$

Whakarūnātia te whārite tuarua ki te āhua tānga ngahuru.

$$4\left(x+8\right)+40y-26=0$$

Whakareatia 4 ki te x+8.

$$4x+32+40y-26=0$$

Tāpiri 32 ki te -26.

$$4x+40y+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$4x+40y=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-2\\4&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\4&40\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}4&-2\\4&40\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-2\\4&40\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{40}{4\times 40-\left(-2\times 4\right)}&-\frac{-2}{4\times 40-\left(-2\times 4\right)}\\-\frac{4}{4\times 40-\left(-2\times 4\right)}&\frac{4}{4\times 40-\left(-2\times 4\right)}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}&\frac{1}{84}\\-\frac{1}{42}&\frac{1}{42}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{21}\times 6+\frac{1}{84}\left(-6\right)\\-\frac{1}{42}\times 6+\frac{1}{42}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{19}{14}\\-\frac{2}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{19}{14},y=-\frac{2}{7}$$