Whakaoti i te {l}{2x-y=4}{3x-5y=15}

$$\left\{ \begin{array} { l } { 2 x - y = 4 } \\ { 3 x - 5 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{5}{7}\approx 0.714285714$$

$$y = -\frac{18}{7} = -2\frac{4}{7} \approx -2.571428571$$

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.

$$2x-y=4,3x-5y=15$$

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

$$2x-y=4$$

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

$$2x=y+4$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Whakakapia te \frac{y}{2}+2 mō te x ki tērā atu whārite, 3x-5y=15.

$$3\left(\frac{1}{2}y+2\right)-5y=15$$

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

$$\frac{3}{2}y+6-5y=15$$

Tāpiri \frac{3y}{2} ki te -5y.

$$-\frac{7}{2}y+6=15$$

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

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

Whakawehea ngā taha e rua o te whārite ki te -\frac{7}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{18}{7}$$

Whakaurua te -\frac{18}{7} mō y ki x=\frac{1}{2}y+2. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-\frac{18}{7}\right)+2$$

Whakareatia \frac{1}{2} ki te -\frac{18}{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.

$$x=-\frac{9}{7}+2$$

Tāpiri 2 ki te -\frac{9}{7}.

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

Kua oti te pūnaha te whakatau.

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

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.

$$2x-y=4,3x-5y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\3&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right))\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right))\left(\begin{matrix}4\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right))\left(\begin{matrix}4\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\3&-5\end{matrix}\right))\left(\begin{matrix}4\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{2\left(-5\right)-\left(-3\right)}&-\frac{-1}{2\left(-5\right)-\left(-3\right)}\\-\frac{3}{2\left(-5\right)-\left(-3\right)}&\frac{2}{2\left(-5\right)-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}4\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}&-\frac{1}{7}\\\frac{3}{7}&-\frac{2}{7}\end{matrix}\right)\left(\begin{matrix}4\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{5}{7}\times 4-\frac{1}{7}\times 15\\\frac{3}{7}\times 4-\frac{2}{7}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

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

Ngā Upane e Whakamahi i te Tangohanga

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.

$$2x-y=4,3x-5y=15$$

Kia ōrite ai a 2x 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 2.

$$3\times 2x+3\left(-1\right)y=3\times 4,2\times 3x+2\left(-5\right)y=2\times 15$$

Whakarūnātia.

$$6x-3y=12,6x-10y=30$$

Me tango 6x-10y=30 mai i 6x-3y=12 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$6x-6x-3y+10y=12-30$$

Tāpiri 6x ki te -6x. Ka whakakore atu ngā kupu 6x me -6x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-3y+10y=12-30$$

Tāpiri -3y ki te 10y.

$$7y=12-30$$

Tāpiri 12 ki te -30.

$$7y=-18$$

Whakawehea ngā taha e rua ki te 7.

$$y=-\frac{18}{7}$$

Whakaurua te -\frac{18}{7} mō y ki 3x-5y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-5\left(-\frac{18}{7}\right)=15$$

Whakareatia -5 ki te -\frac{18}{7}.

$$3x+\frac{90}{7}=15$$

Me tango \frac{90}{7} mai i ngā taha e rua o te whārite.

$$3x=\frac{15}{7}$$

Whakawehea ngā taha e rua ki te 3.

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

Kua oti te pūnaha te whakatau.

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