Whakaoti i te {l}{-7x+2y=-24}{5x-y=18}

$$\left. \begin{array} { l } { - 7 x + 2 y = - 24 } \\ { 5 x - y = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=2$$

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.

$$-7x+2y=-24,5x-y=18$$

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

$$-7x+2y=-24$$

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

$$-7x=-2y-24$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{1}{7}\left(-2y-24\right)$$

Whakareatia -\frac{1}{7} ki te -2y-24.

$$x=\frac{2}{7}y+\frac{24}{7}$$

Whakakapia te \frac{24+2y}{7} mō te x ki tērā atu whārite, 5x-y=18.

$$5\left(\frac{2}{7}y+\frac{24}{7}\right)-y=18$$

Whakareatia 5 ki te \frac{24+2y}{7}.

$$\frac{10}{7}y+\frac{120}{7}-y=18$$

Tāpiri \frac{10y}{7} ki te -y.

$$\frac{3}{7}y+\frac{120}{7}=18$$

Me tango \frac{120}{7} mai i ngā taha e rua o te whārite.

$$\frac{3}{7}y=\frac{6}{7}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=2$$

Whakaurua te 2 mō y ki x=\frac{2}{7}y+\frac{24}{7}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{2}{7}\times 2+\frac{24}{7}$$

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

$$x=\frac{4+24}{7}$$

Tāpiri \frac{24}{7} ki te \frac{4}{7} 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.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$

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.

$$-7x+2y=-24,5x-y=18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-24\\18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-7&2\\5&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right))\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right))\left(\begin{matrix}-24\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right))\left(\begin{matrix}-24\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-7&2\\5&-1\end{matrix}\right))\left(\begin{matrix}-24\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-7\left(-1\right)-2\times 5}&-\frac{2}{-7\left(-1\right)-2\times 5}\\-\frac{5}{-7\left(-1\right)-2\times 5}&-\frac{7}{-7\left(-1\right)-2\times 5}\end{matrix}\right)\left(\begin{matrix}-24\\18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&\frac{2}{3}\\\frac{5}{3}&\frac{7}{3}\end{matrix}\right)\left(\begin{matrix}-24\\18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\left(-24\right)+\frac{2}{3}\times 18\\\frac{5}{3}\left(-24\right)+\frac{7}{3}\times 18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=2$$

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.

$$-7x+2y=-24,5x-y=18$$

Kia ōrite ai a -7x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -7.

$$5\left(-7\right)x+5\times 2y=5\left(-24\right),-7\times 5x-7\left(-1\right)y=-7\times 18$$

Whakarūnātia.

$$-35x+10y=-120,-35x+7y=-126$$

Me tango -35x+7y=-126 mai i -35x+10y=-120 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-35x+35x+10y-7y=-120+126$$

Tāpiri -35x ki te 35x. Ka whakakore atu ngā kupu -35x me 35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$10y-7y=-120+126$$

Tāpiri 10y ki te -7y.

$$3y=-120+126$$

Tāpiri -120 ki te 126.

$$3y=6$$

Whakawehea ngā taha e rua ki te 3.

$$y=2$$

Whakaurua te 2 mō y ki 5x-y=18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-2=18$$

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

$$5x=20$$

Whakawehea ngā taha e rua ki te 5.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=2$$