Whakaoti i te {l}{3x+y=5}{7x+y=6}

$$\left. \begin{array} { l } { 3 x + y = 5 } \\ { 7 x + y = 6 } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{1}{4}=0.25$$

$$y = \frac{17}{4} = 4\frac{1}{4} = 4.25$$

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.

$$3x+y=5,7x+y=6$$

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

$$3x+y=5$$

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

$$3x=-y+5$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=-\frac{1}{3}y+\frac{5}{3}$$

Whakakapia te \frac{-y+5}{3} mō te x ki tērā atu whārite, 7x+y=6.

$$7\left(-\frac{1}{3}y+\frac{5}{3}\right)+y=6$$

Whakareatia 7 ki te \frac{-y+5}{3}.

$$-\frac{7}{3}y+\frac{35}{3}+y=6$$

Tāpiri -\frac{7y}{3} ki te y.

$$-\frac{4}{3}y+\frac{35}{3}=6$$

Me tango \frac{35}{3} mai i ngā taha e rua o te whārite.

$$-\frac{4}{3}y=-\frac{17}{3}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{4}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{17}{4}$$

Whakaurua te \frac{17}{4} mō y ki x=-\frac{1}{3}y+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{3}\times \frac{17}{4}+\frac{5}{3}$$

Whakareatia -\frac{1}{3} ki te \frac{17}{4} 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.

$$x=-\frac{17}{12}+\frac{5}{3}$$

Tāpiri \frac{5}{3} ki te -\frac{17}{12} 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.

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

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{17}{4}$$

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.

$$3x+y=5,7x+y=6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&1\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\7&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&1\\7&1\end{matrix}\right))\left(\begin{matrix}3&1\\7&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\7&1\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\7&1\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&1\\7&1\end{matrix}\right))\left(\begin{matrix}5\\6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3-7}&-\frac{1}{3-7}\\-\frac{7}{3-7}&\frac{3}{3-7}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}&\frac{1}{4}\\\frac{7}{4}&-\frac{3}{4}\end{matrix}\right)\left(\begin{matrix}5\\6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4}\times 5+\frac{1}{4}\times 6\\\frac{7}{4}\times 5-\frac{3}{4}\times 6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4}\\\frac{17}{4}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{1}{4},y=\frac{17}{4}$$

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.

$$3x+y=5,7x+y=6$$

Me tango 7x+y=6 mai i 3x+y=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-7x+y-y=5-6$$

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

$$3x-7x=5-6$$

Tāpiri 3x ki te -7x.

$$-4x=5-6$$

Tāpiri 5 ki te -6.

$$-4x=-1$$

Whakawehea ngā taha e rua ki te -4.

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

Whakaurua te \frac{1}{4} mō x ki 7x+y=6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$7\times \frac{1}{4}+y=6$$

Whakareatia 7 ki te \frac{1}{4}.

$$\frac{7}{4}+y=6$$

Me tango \frac{7}{4} mai i ngā taha e rua o te whārite.

$$y=\frac{17}{4}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{1}{4},y=\frac{17}{4}$$