Whakaoti i te {l}{3x-5y-4=0}{15y=4x+3}

$$\left\{ \begin{array} { l } { 3 x - 5 y - 4 = 0 } \\ { 15 y = 4 x + 3 } \end{array} \right.$$

Whakaoti mō x, y

$$x=3$$

$$y=1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te 4 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-5y=4$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$15y-4x=3$$

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.

$$3x-5y=4,-4x+15y=3$$

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

$$3x-5y=4$$

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

$$3x=5y+4$$

Whakawehea ngā taha e rua ki te 3.

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

Whakareatia \frac{1}{3} ki te 5y+4.

$$x=\frac{5}{3}y+\frac{4}{3}$$

Whakakapia te \frac{5y+4}{3} mō te x ki tērā atu whārite, -4x+15y=3.

$$-4\left(\frac{5}{3}y+\frac{4}{3}\right)+15y=3$$

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

$$-\frac{20}{3}y-\frac{16}{3}+15y=3$$

Tāpiri -\frac{20y}{3} ki te 15y.

$$\frac{25}{3}y-\frac{16}{3}=3$$

Me tāpiri \frac{16}{3} ki ngā taha e rua o te whārite.

$$\frac{25}{3}y=\frac{25}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{25}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=1$$

Whakaurua te 1 mō y ki x=\frac{5}{3}y+\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5+4}{3}$$

Tāpiri \frac{4}{3} ki te \frac{5}{3} 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.

$$x=3,y=1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te 4 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-5y=4$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$15y-4x=3$$

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$3x-5y=4,-4x+15y=3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-5\\-4&15\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right))\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-5\\-4&15\end{matrix}\right))\left(\begin{matrix}4\\3\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{15}{3\times 15-\left(-5\left(-4\right)\right)}&-\frac{-5}{3\times 15-\left(-5\left(-4\right)\right)}\\-\frac{-4}{3\times 15-\left(-5\left(-4\right)\right)}&\frac{3}{3\times 15-\left(-5\left(-4\right)\right)}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}&\frac{1}{5}\\\frac{4}{25}&\frac{3}{25}\end{matrix}\right)\left(\begin{matrix}4\\3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{5}\times 4+\frac{1}{5}\times 3\\\frac{4}{25}\times 4+\frac{3}{25}\times 3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\1\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=3,y=1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te 4 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x-5y=4$$

Whakaarohia te whārite tuarua. Tangohia te 4x mai i ngā taha e rua.

$$15y-4x=3$$

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.

$$3x-5y=4,-4x+15y=3$$

Kia ōrite ai a 3x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 3.

$$-4\times 3x-4\left(-5\right)y=-4\times 4,3\left(-4\right)x+3\times 15y=3\times 3$$

Whakarūnātia.

$$-12x+20y=-16,-12x+45y=9$$

Me tango -12x+45y=9 mai i -12x+20y=-16 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-12x+12x+20y-45y=-16-9$$

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.

$$20y-45y=-16-9$$

Tāpiri 20y ki te -45y.

$$-25y=-16-9$$

Tāpiri -16 ki te -9.

$$-25y=-25$$

Whakawehea ngā taha e rua ki te -25.

$$y=1$$

Whakaurua te 1 mō y ki -4x+15y=3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x+15=3$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-4x=-12$$

Whakawehea ngā taha e rua ki te -4.

$$x=3$$

Kua oti te pūnaha te whakatau.

$$x=3,y=1$$