Whakaoti i te {l}{4x-5y=9}{7x-4y=15}

$$\left\{ \begin{array} { l } { 4 x - 5 y = 9 } \\ { 7 x - 4 y = 15 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{39}{19} = 2\frac{1}{19} \approx 2.052631579$$

$$y=-\frac{3}{19}\approx -0.157894737$$

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.

$$4x-5y=9,7x-4y=15$$

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

$$4x-5y=9$$

Me tāpiri 5y ki ngā taha e rua o te whārite.

$$4x=5y+9$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(5y+9\right)$$

Whakareatia \frac{1}{4} ki te 5y+9.

$$x=\frac{5}{4}y+\frac{9}{4}$$

Whakakapia te \frac{5y+9}{4} mō te x ki tērā atu whārite, 7x-4y=15.

$$7\left(\frac{5}{4}y+\frac{9}{4}\right)-4y=15$$

Whakareatia 7 ki te \frac{5y+9}{4}.

$$\frac{35}{4}y+\frac{63}{4}-4y=15$$

Tāpiri \frac{35y}{4} ki te -4y.

$$\frac{19}{4}y+\frac{63}{4}=15$$

Me tango \frac{63}{4} mai i ngā taha e rua o te whārite.

$$\frac{19}{4}y=-\frac{3}{4}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-\frac{3}{19}$$

Whakaurua te -\frac{3}{19} mō y ki x=\frac{5}{4}y+\frac{9}{4}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4}\left(-\frac{3}{19}\right)+\frac{9}{4}$$

Whakareatia \frac{5}{4} ki te -\frac{3}{19} 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.

$$x=-\frac{15}{76}+\frac{9}{4}$$

Tāpiri \frac{9}{4} ki te -\frac{15}{76} 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.

$$x=\frac{39}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{39}{19},y=-\frac{3}{19}$$

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.

$$4x-5y=9,7x-4y=15$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}9\\15\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-5\\7&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right))\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&-5\\7&-4\end{matrix}\right))\left(\begin{matrix}9\\15\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{4\left(-4\right)-\left(-5\times 7\right)}&-\frac{-5}{4\left(-4\right)-\left(-5\times 7\right)}\\-\frac{7}{4\left(-4\right)-\left(-5\times 7\right)}&\frac{4}{4\left(-4\right)-\left(-5\times 7\right)}\end{matrix}\right)\left(\begin{matrix}9\\15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{19}&\frac{5}{19}\\-\frac{7}{19}&\frac{4}{19}\end{matrix}\right)\left(\begin{matrix}9\\15\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{19}\times 9+\frac{5}{19}\times 15\\-\frac{7}{19}\times 9+\frac{4}{19}\times 15\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{39}{19}\\-\frac{3}{19}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{39}{19},y=-\frac{3}{19}$$

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.

$$4x-5y=9,7x-4y=15$$

Kia ōrite ai a 4x me 7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$7\times 4x+7\left(-5\right)y=7\times 9,4\times 7x+4\left(-4\right)y=4\times 15$$

Whakarūnātia.

$$28x-35y=63,28x-16y=60$$

Me tango 28x-16y=60 mai i 28x-35y=63 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$28x-28x-35y+16y=63-60$$

Tāpiri 28x ki te -28x. Ka whakakore atu ngā kupu 28x me -28x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-35y+16y=63-60$$

Tāpiri -35y ki te 16y.

$$-19y=63-60$$

Tāpiri 63 ki te -60.

$$-19y=3$$

Whakawehea ngā taha e rua ki te -19.

$$y=-\frac{3}{19}$$

Whakaurua te -\frac{3}{19} mō y ki 7x-4y=15. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$7x-4\left(-\frac{3}{19}\right)=15$$

Whakareatia -4 ki te -\frac{3}{19}.

$$7x+\frac{12}{19}=15$$

Me tango \frac{12}{19} mai i ngā taha e rua o te whārite.

$$7x=\frac{273}{19}$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{39}{19}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{39}{19},y=-\frac{3}{19}$$