Whakaoti i te {l}{x+2y=-18}{3x-y=-1}

$$\left\{ \begin{array} { l } { x + 2 y = - 18 } \\ { 3 x - y = - 1 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{20}{7} = -2\frac{6}{7} \approx -2.857142857$$

$$y = -\frac{53}{7} = -7\frac{4}{7} \approx -7.571428571$$

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.

$$x+2y=-18,3x-y=-1$$

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

$$x+2y=-18$$

Me tango 2y mai i ngā taha e rua o te whārite.

$$x=-2y-18$$

Whakakapia te -2y-18 mō te x ki tērā atu whārite, 3x-y=-1.

$$3\left(-2y-18\right)-y=-1$$

Whakareatia 3 ki te -2y-18.

$$-6y-54-y=-1$$

Tāpiri -6y ki te -y.

$$-7y-54=-1$$

Me tāpiri 54 ki ngā taha e rua o te whārite.

$$-7y=53$$

Whakawehea ngā taha e rua ki te -7.

$$y=-\frac{53}{7}$$

Whakaurua te -\frac{53}{7} mō y ki x=-2y-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-2\left(-\frac{53}{7}\right)-18$$

Whakareatia -2 ki te -\frac{53}{7}.

$$x=\frac{106}{7}-18$$

Tāpiri -18 ki te \frac{106}{7}.

$$x=-\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{20}{7},y=-\frac{53}{7}$$

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.

$$x+2y=-18,3x-y=-1$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}1&2\\3&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&2\\3&-1\end{matrix}\right))\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-2\times 3}&-\frac{2}{-1-2\times 3}\\-\frac{3}{-1-2\times 3}&\frac{1}{-1-2\times 3}\end{matrix}\right)\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&\frac{2}{7}\\\frac{3}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-18\\-1\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\left(-18\right)+\frac{2}{7}\left(-1\right)\\\frac{3}{7}\left(-18\right)-\frac{1}{7}\left(-1\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{20}{7}\\-\frac{53}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{20}{7},y=-\frac{53}{7}$$

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.

$$x+2y=-18,3x-y=-1$$

Kia ōrite ai a x 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 1.

$$3x+3\times 2y=3\left(-18\right),3x-y=-1$$

Whakarūnātia.

$$3x+6y=-54,3x-y=-1$$

Me tango 3x-y=-1 mai i 3x+6y=-54 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+6y+y=-54+1$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y+y=-54+1$$

Tāpiri 6y ki te y.

$$7y=-54+1$$

Tāpiri -54 ki te 1.

$$7y=-53$$

Whakawehea ngā taha e rua ki te 7.

$$y=-\frac{53}{7}$$

Whakaurua te -\frac{53}{7} mō y ki 3x-y=-1. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-\left(-\frac{53}{7}\right)=-1$$

Me tango \frac{53}{7} mai i ngā taha e rua o te whārite.

$$3x=-\frac{60}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{20}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{20}{7},y=-\frac{53}{7}$$