Whakaoti i te {l}{4x+5y=10}{3x-3y=21}

$$\left. \begin{array} { l } { 4 x + 5 y = 10 } \\ { 3 x - 3 y = 21 } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-2$$

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.

$$4x+5y=10,3x-3y=21$$

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

$$4x+5y=10$$

Me tango 5y mai i ngā taha e rua o te whārite.

$$4x=-5y+10$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y+10\right)$$

Whakareatia \frac{1}{4} ki te -5y+10.

$$x=-\frac{5}{4}y+\frac{5}{2}$$

Whakakapia te -\frac{5y}{4}+\frac{5}{2} mō te x ki tērā atu whārite, 3x-3y=21.

$$3\left(-\frac{5}{4}y+\frac{5}{2}\right)-3y=21$$

Whakareatia 3 ki te -\frac{5y}{4}+\frac{5}{2}.

$$-\frac{15}{4}y+\frac{15}{2}-3y=21$$

Tāpiri -\frac{15y}{4} ki te -3y.

$$-\frac{27}{4}y+\frac{15}{2}=21$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$-\frac{27}{4}y=\frac{27}{2}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{27}{4}, 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{5}{4}y+\frac{5}{2}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\left(-2\right)+\frac{5}{2}$$

Whakareatia -\frac{5}{4} ki te -2.

$$x=\frac{5+5}{2}$$

Tāpiri \frac{5}{2} ki te \frac{5}{2} 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.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$

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.

$$4x+5y=10,3x-3y=21$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}10\\21\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\3&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\3&-3\end{matrix}\right))\left(\begin{matrix}4&5\\3&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\3&-3\end{matrix}\right))\left(\begin{matrix}10\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\3&-3\end{matrix}\right))\left(\begin{matrix}10\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\3&-3\end{matrix}\right))\left(\begin{matrix}10\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{4\left(-3\right)-5\times 3}&-\frac{5}{4\left(-3\right)-5\times 3}\\-\frac{3}{4\left(-3\right)-5\times 3}&\frac{4}{4\left(-3\right)-5\times 3}\end{matrix}\right)\left(\begin{matrix}10\\21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}&\frac{5}{27}\\\frac{1}{9}&-\frac{4}{27}\end{matrix}\right)\left(\begin{matrix}10\\21\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{9}\times 10+\frac{5}{27}\times 21\\\frac{1}{9}\times 10-\frac{4}{27}\times 21\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\-2\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-2$$

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.

$$4x+5y=10,3x-3y=21$$

Kia ōrite ai a 4x 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 4.

$$3\times 4x+3\times 5y=3\times 10,4\times 3x+4\left(-3\right)y=4\times 21$$

Whakarūnātia.

$$12x+15y=30,12x-12y=84$$

Me tango 12x-12y=84 mai i 12x+15y=30 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x+15y+12y=30-84$$

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.

$$15y+12y=30-84$$

Tāpiri 15y ki te 12y.

$$27y=30-84$$

Tāpiri 30 ki te -84.

$$27y=-54$$

Whakawehea ngā taha e rua ki te 27.

$$y=-2$$

Whakaurua te -2 mō y ki 3x-3y=21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x-3\left(-2\right)=21$$

Whakareatia -3 ki te -2.

$$3x+6=21$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x=15$$

Whakawehea ngā taha e rua ki te 3.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-2$$