Whakaoti i te {l}{2x+3y=30}{6x+8y=42}

$$\left. \begin{array} { l } { 2 x + 3 y = 30 } \\ { 6 x + 8 y = 42 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-57$$

$$y=48$$

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.

$$2x+3y=30,6x+8y=42$$

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

$$2x+3y=30$$

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

$$2x=-3y+30$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(-3y+30\right)$$

Whakareatia \frac{1}{2} ki te -3y+30.

$$x=-\frac{3}{2}y+15$$

Whakakapia te -\frac{3y}{2}+15 mō te x ki tērā atu whārite, 6x+8y=42.

$$6\left(-\frac{3}{2}y+15\right)+8y=42$$

Whakareatia 6 ki te -\frac{3y}{2}+15.

$$-9y+90+8y=42$$

Tāpiri -9y ki te 8y.

$$-y+90=42$$

Me tango 90 mai i ngā taha e rua o te whārite.

$$-y=-48$$

Whakawehea ngā taha e rua ki te -1.

$$y=48$$

Whakaurua te 48 mō y ki x=-\frac{3}{2}y+15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{2}\times 48+15$$

Whakareatia -\frac{3}{2} ki te 48.

$$x=-72+15$$

Tāpiri 15 ki te -72.

$$x=-57$$

Kua oti te pūnaha te whakatau.

$$x=-57,y=48$$

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.

$$2x+3y=30,6x+8y=42$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&3\\6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}30\\42\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&3\\6&8\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\6&8\end{matrix}\right))\left(\begin{matrix}2&3\\6&8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&8\end{matrix}\right))\left(\begin{matrix}30\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&8\end{matrix}\right))\left(\begin{matrix}30\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\6&8\end{matrix}\right))\left(\begin{matrix}30\\42\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{2\times 8-3\times 6}&-\frac{3}{2\times 8-3\times 6}\\-\frac{6}{2\times 8-3\times 6}&\frac{2}{2\times 8-3\times 6}\end{matrix}\right)\left(\begin{matrix}30\\42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4&\frac{3}{2}\\3&-1\end{matrix}\right)\left(\begin{matrix}30\\42\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\times 30+\frac{3}{2}\times 42\\3\times 30-42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-57\\48\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-57,y=48$$

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.

$$2x+3y=30,6x+8y=42$$

Kia ōrite ai a 2x me 6x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 6 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$6\times 2x+6\times 3y=6\times 30,2\times 6x+2\times 8y=2\times 42$$

Whakarūnātia.

$$12x+18y=180,12x+16y=84$$

Me tango 12x+16y=84 mai i 12x+18y=180 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+18y-16y=180-84$$

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.

$$18y-16y=180-84$$

Tāpiri 18y ki te -16y.

$$2y=180-84$$

Tāpiri 180 ki te -84.

$$2y=96$$

Whakawehea ngā taha e rua ki te 2.

$$y=48$$

Whakaurua te 48 mō y ki 6x+8y=42. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$6x+8\times 48=42$$

Whakareatia 8 ki te 48.

$$6x+384=42$$

Me tango 384 mai i ngā taha e rua o te whārite.

$$6x=-342$$

Whakawehea ngā taha e rua ki te 6.

$$x=-57$$

Kua oti te pūnaha te whakatau.

$$x=-57,y=48$$