Whakaoti i te {l}{8x+3y=5}{3x+2y=70}

$$\left. \begin{array} { l } { 8 x + 3 y = 5 } \\ { 3 x + 2 y = 70 } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{200}{7} = -28\frac{4}{7} \approx -28.571428571$$

$$y = \frac{545}{7} = 77\frac{6}{7} \approx 77.857142857$$

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.

$$8x+3y=5,3x+2y=70$$

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

$$8x+3y=5$$

Me tango 3y mai i ngā taha e rua o te whārite.

$$8x=-3y+5$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-3y+5\right)$$

Whakareatia \frac{1}{8} ki te -3y+5.

$$x=-\frac{3}{8}y+\frac{5}{8}$$

Whakakapia te \frac{-3y+5}{8} mō te x ki tērā atu whārite, 3x+2y=70.

$$3\left(-\frac{3}{8}y+\frac{5}{8}\right)+2y=70$$

Whakareatia 3 ki te \frac{-3y+5}{8}.

$$-\frac{9}{8}y+\frac{15}{8}+2y=70$$

Tāpiri -\frac{9y}{8} ki te 2y.

$$\frac{7}{8}y+\frac{15}{8}=70$$

Me tango \frac{15}{8} mai i ngā taha e rua o te whārite.

$$\frac{7}{8}y=\frac{545}{8}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{545}{7}$$

Whakaurua te \frac{545}{7} mō y ki x=-\frac{3}{8}y+\frac{5}{8}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{8}\times \frac{545}{7}+\frac{5}{8}$$

Whakareatia -\frac{3}{8} ki te \frac{545}{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.

$$x=-\frac{1635}{56}+\frac{5}{8}$$

Tāpiri \frac{5}{8} ki te -\frac{1635}{56} 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.

$$x=-\frac{200}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{200}{7},y=\frac{545}{7}$$

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.

$$8x+3y=5,3x+2y=70$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&3\\3&2\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&3\\3&2\end{matrix}\right))\left(\begin{matrix}8&3\\3&2\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&3\\3&2\end{matrix}\right))\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{8\times 2-3\times 3}&-\frac{3}{8\times 2-3\times 3}\\-\frac{3}{8\times 2-3\times 3}&\frac{8}{8\times 2-3\times 3}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&-\frac{3}{7}\\-\frac{3}{7}&\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}5\\70\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 5-\frac{3}{7}\times 70\\-\frac{3}{7}\times 5+\frac{8}{7}\times 70\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{200}{7}\\\frac{545}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{200}{7},y=\frac{545}{7}$$

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.

$$8x+3y=5,3x+2y=70$$

Kia ōrite ai a 8x 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 8.

$$3\times 8x+3\times 3y=3\times 5,8\times 3x+8\times 2y=8\times 70$$

Whakarūnātia.

$$24x+9y=15,24x+16y=560$$

Me tango 24x+16y=560 mai i 24x+9y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$24x-24x+9y-16y=15-560$$

Tāpiri 24x ki te -24x. Ka whakakore atu ngā kupu 24x me -24x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$9y-16y=15-560$$

Tāpiri 9y ki te -16y.

$$-7y=15-560$$

Tāpiri 15 ki te -560.

$$-7y=-545$$

Whakawehea ngā taha e rua ki te -7.

$$y=\frac{545}{7}$$

Whakaurua te \frac{545}{7} mō y ki 3x+2y=70. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\times \frac{545}{7}=70$$

Whakareatia 2 ki te \frac{545}{7}.

$$3x+\frac{1090}{7}=70$$

Me tango \frac{1090}{7} mai i ngā taha e rua o te whārite.

$$3x=-\frac{600}{7}$$

Whakawehea ngā taha e rua ki te 3.

$$x=-\frac{200}{7}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{200}{7},y=\frac{545}{7}$$