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

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

Whakaoti mō x, y

$$x=-1$$

$$y = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

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.

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

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

$$4x-2y=5$$

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

$$4x=2y+5$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(2y+5\right)$$

Whakareatia \frac{1}{4} ki te 2y+5.

$$x=\frac{1}{2}y+\frac{5}{4}$$

Whakakapia te \frac{y}{2}+\frac{5}{4} mō te x ki tērā atu whārite, 3x-4y=15.

$$3\left(\frac{1}{2}y+\frac{5}{4}\right)-4y=15$$

Whakareatia 3 ki te \frac{y}{2}+\frac{5}{4}.

$$\frac{3}{2}y+\frac{15}{4}-4y=15$$

Tāpiri \frac{3y}{2} ki te -4y.

$$-\frac{5}{2}y+\frac{15}{4}=15$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$-\frac{5}{2}y=\frac{45}{4}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

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

Whakaurua te -\frac{9}{2} mō y ki x=\frac{1}{2}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-\frac{9}{2}\right)+\frac{5}{4}$$

Whakareatia \frac{1}{2} ki te -\frac{9}{2} 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.

$$x=\frac{-9+5}{4}$$

Tāpiri \frac{5}{4} ki te -\frac{9}{4} 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.

$$x=-1,y=-\frac{9}{2}$$

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.

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-2\\3&-4\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}&-\frac{1}{5}\\\frac{3}{10}&-\frac{2}{5}\end{matrix}\right)\left(\begin{matrix}5\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{5}\times 5-\frac{1}{5}\times 15\\\frac{3}{10}\times 5-\frac{2}{5}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-\frac{9}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-1,y=-\frac{9}{2}$$

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.

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

Kia ōrite ai a 4x 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 4.

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

Whakarūnātia.

$$12x-6y=15,12x-16y=60$$

Me tango 12x-16y=60 mai i 12x-6y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-6y+16y=15-60$$

Tāpiri 12x ki te -12x. Ka whakakore atu ngā kupu 12x me -12x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-6y+16y=15-60$$

Tāpiri -6y ki te 16y.

$$10y=15-60$$

Tāpiri 15 ki te -60.

$$10y=-45$$

Whakawehea ngā taha e rua ki te 10.

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

Whakaurua te -\frac{9}{2} mō y ki 3x-4y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-4\left(-\frac{9}{2}\right)=15$$

Whakareatia -4 ki te -\frac{9}{2}.

$$3x+18=15$$

Me tango 18 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.

$$x=-1,y=-\frac{9}{2}$$