Whakaoti i te {l}{8x-5y-3=0}{5*(y-3x)=-10}

$$\left. \begin{array} { l } { 8 x - 5 y - 3 = 0 } \\ { 5 \cdot ( y - 3 x ) = - 10 } \end{array} \right.$$

Whakaoti mō x, y

$$x=1$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$8x-5y=3$$

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 5.

$$y-3x=\frac{-10}{5}$$

Whakawehea te -10 ki te 5, kia riro ko -2.

$$y-3x=-2$$

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.

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

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

$$8x-5y=3$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$8x=5y+3$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(5y+3\right)$$

Whakareatia \frac{1}{8} ki te 5y+3.

$$x=\frac{5}{8}y+\frac{3}{8}$$

Whakakapia te \frac{5y+3}{8} mō te x ki tērā atu whārite, -3x+y=-2.

$$-3\left(\frac{5}{8}y+\frac{3}{8}\right)+y=-2$$

Whakareatia -3 ki te \frac{5y+3}{8}.

$$-\frac{15}{8}y-\frac{9}{8}+y=-2$$

Tāpiri -\frac{15y}{8} ki te y.

$$-\frac{7}{8}y-\frac{9}{8}=-2$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

$$-\frac{7}{8}y=-\frac{7}{8}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{5}{8}y+\frac{3}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5+3}{8}$$

Tāpiri \frac{3}{8} ki te \frac{5}{8} 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.

$$x=1,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$8x-5y=3$$

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 5.

$$y-3x=\frac{-10}{5}$$

Whakawehea te -10 ki te 5, kia riro ko -2.

$$y-3x=-2$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&-5\\-3&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}&-\frac{5}{7}\\-\frac{3}{7}&-\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}3\\-2\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{7}\times 3-\frac{5}{7}\left(-2\right)\\-\frac{3}{7}\times 3-\frac{8}{7}\left(-2\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=1,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$8x-5y=3$$

Whakaarohia te whārite tuarua. Whakawehea ngā taha e rua ki te 5.

$$y-3x=\frac{-10}{5}$$

Whakawehea te -10 ki te 5, kia riro ko -2.

$$y-3x=-2$$

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.

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

Kia ōrite ai a 8x me -3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 8.

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

Whakarūnātia.

$$-24x+15y=-9,-24x+8y=-16$$

Me tango -24x+8y=-16 mai i -24x+15y=-9 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-24x+24x+15y-8y=-9+16$$

Tāpiri -24x ki te 24x. Ka whakakore atu ngā kupu -24x me 24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$15y-8y=-9+16$$

Tāpiri 15y ki te -8y.

$$7y=-9+16$$

Tāpiri -9 ki te 16.

$$7y=7$$

Whakawehea ngā taha e rua ki te 7.

$$y=1$$

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

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

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

$$-3x=-3$$

Whakawehea ngā taha e rua ki te -3.

$$x=1$$

Kua oti te pūnaha te whakatau.

$$x=1,y=1$$