Whakaoti i te {r}{4x-y=1}{3x+y=9}

$$\left. \begin{array} { r } { 4 x - y = 1 } \\ { 3 x + y = 9 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{10}{7} = 1\frac{3}{7} \approx 1.428571429$$

$$y = \frac{33}{7} = 4\frac{5}{7} \approx 4.714285714$$

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.

$$4x-y=1,3x+y=9$$

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

$$4x-y=1$$

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

$$4x=y+1$$

Whakawehea ngā taha e rua ki te 4.

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

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

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

Whakakapia te \frac{1+y}{4} mō te x ki tērā atu whārite, 3x+y=9.

$$3\left(\frac{1}{4}y+\frac{1}{4}\right)+y=9$$

Whakareatia 3 ki te \frac{1+y}{4}.

$$\frac{3}{4}y+\frac{3}{4}+y=9$$

Tāpiri \frac{3y}{4} ki te y.

$$\frac{7}{4}y+\frac{3}{4}=9$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

$$\frac{7}{4}y=\frac{33}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{33}{7}$$

Whakaurua te \frac{33}{7} mō y ki x=\frac{1}{4}y+\frac{1}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\times \frac{33}{7}+\frac{1}{4}$$

Whakareatia \frac{1}{4} ki te \frac{33}{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.

$$x=\frac{33}{28}+\frac{1}{4}$$

Tāpiri \frac{1}{4} ki te \frac{33}{28} 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.

$$x=\frac{10}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10}{7},y=\frac{33}{7}$$

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.

$$4x-y=1,3x+y=9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\9\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\3&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\3&1\end{matrix}\right))\left(\begin{matrix}4&-1\\3&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&1\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&1\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&1\end{matrix}\right))\left(\begin{matrix}1\\9\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4-\left(-3\right)}&-\frac{-1}{4-\left(-3\right)}\\-\frac{3}{4-\left(-3\right)}&\frac{4}{4-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}1\\9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{1}{7}\\-\frac{3}{7}&\frac{4}{7}\end{matrix}\right)\left(\begin{matrix}1\\9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}+\frac{1}{7}\times 9\\-\frac{3}{7}+\frac{4}{7}\times 9\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{7}\\\frac{33}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{10}{7},y=\frac{33}{7}$$

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.

$$4x-y=1,3x+y=9$$

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.

$$3\times 4x+3\left(-1\right)y=3,4\times 3x+4y=4\times 9$$

Whakarūnātia.

$$12x-3y=3,12x+4y=36$$

Me tango 12x+4y=36 mai i 12x-3y=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-3y-4y=3-36$$

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.

$$-3y-4y=3-36$$

Tāpiri -3y ki te -4y.

$$-7y=3-36$$

Tāpiri 3 ki te -36.

$$-7y=-33$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{33}{7}$$

Whakaurua te \frac{33}{7} mō y ki 3x+y=9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+\frac{33}{7}=9$$

Me tango \frac{33}{7} mai i ngā taha e rua o te whārite.

$$3x=\frac{30}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{10}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{10}{7},y=\frac{33}{7}$$