Whakaoti i te {l}{4x-y=5}{3x+3y=30}

$$\left. \begin{array} { l } { 4 x - y = 5 } \\ { 3 x + 3 y = 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=7$$

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.

$$4x-y=5,3x+3y=30$$

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

$$4x-y=5$$

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

$$4x=y+5$$

Whakawehea ngā taha e rua ki te 4.

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

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

$$x=\frac{1}{4}y+\frac{5}{4}$$

Whakakapia te \frac{5+y}{4} mō te x ki tērā atu whārite, 3x+3y=30.

$$3\left(\frac{1}{4}y+\frac{5}{4}\right)+3y=30$$

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

$$\frac{3}{4}y+\frac{15}{4}+3y=30$$

Tāpiri \frac{3y}{4} ki te 3y.

$$\frac{15}{4}y+\frac{15}{4}=30$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$\frac{15}{4}y=\frac{105}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{15}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=7$$

Whakaurua te 7 mō y ki x=\frac{1}{4}y+\frac{5}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{4}\times 7+\frac{5}{4}$$

Whakareatia \frac{1}{4} ki te 7.

$$x=\frac{7+5}{4}$$

Tāpiri \frac{5}{4} ki te \frac{7}{4} 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.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=7$$

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.

$$4x-y=5,3x+3y=30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-1\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-1\\3&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-1\\3&3\end{matrix}\right))\left(\begin{matrix}4&-1\\3&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-1\\3&3\end{matrix}\right))\left(\begin{matrix}5\\30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{4\times 3-\left(-3\right)}&-\frac{-1}{4\times 3-\left(-3\right)}\\-\frac{3}{4\times 3-\left(-3\right)}&\frac{4}{4\times 3-\left(-3\right)}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}&\frac{1}{15}\\-\frac{1}{5}&\frac{4}{15}\end{matrix}\right)\left(\begin{matrix}5\\30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{5}\times 5+\frac{1}{15}\times 30\\-\frac{1}{5}\times 5+\frac{4}{15}\times 30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\7\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=7$$

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.

$$4x-y=5,3x+3y=30$$

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.

$$3\times 4x+3\left(-1\right)y=3\times 5,4\times 3x+4\times 3y=4\times 30$$

Whakarūnātia.

$$12x-3y=15,12x+12y=120$$

Me tango 12x+12y=120 mai i 12x-3y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$12x-12x-3y-12y=15-120$$

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.

$$-3y-12y=15-120$$

Tāpiri -3y ki te -12y.

$$-15y=15-120$$

Tāpiri 15 ki te -120.

$$-15y=-105$$

Whakawehea ngā taha e rua ki te -15.

$$y=7$$

Whakaurua te 7 mō y ki 3x+3y=30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+3\times 7=30$$

Whakareatia 3 ki te 7.

$$3x+21=30$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$3x=9$$

Whakawehea ngā taha e rua ki te 3.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=7$$