Whakaoti i te {l}{x/3+y/5=7}{x/6-2y/5=-4}

$$\left. \begin{array} { l } { \frac { x } { 3 } + \frac { y } { 5 } = 7 } \\ { \frac { x } { 6 } - \frac { 2 y } { 5 } = - 4 } \end{array} \right.$$

Whakaoti mō x, y

$$x=12$$

$$y=15$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 3,5.

$$5x+3y=105$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6\times 2y=-120$$

Whakareatia te -6 ki te 2, ka -12.

$$5x-12y=-120$$

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.

$$5x+3y=105,5x-12y=-120$$

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

$$5x+3y=105$$

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

$$5x=-3y+105$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(-3y+105\right)$$

Whakareatia \frac{1}{5} ki te -3y+105.

$$x=-\frac{3}{5}y+21$$

Whakakapia te -\frac{3y}{5}+21 mō te x ki tērā atu whārite, 5x-12y=-120.

$$5\left(-\frac{3}{5}y+21\right)-12y=-120$$

Whakareatia 5 ki te -\frac{3y}{5}+21.

$$-3y+105-12y=-120$$

Tāpiri -3y ki te -12y.

$$-15y+105=-120$$

Me tango 105 mai i ngā taha e rua o te whārite.

$$-15y=-225$$

Whakawehea ngā taha e rua ki te -15.

$$y=15$$

Whakaurua te 15 mō y ki x=-\frac{3}{5}y+21. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{3}{5}\times 15+21$$

Whakareatia -\frac{3}{5} ki te 15.

$$x=-9+21$$

Tāpiri 21 ki te -9.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=15$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 3,5.

$$5x+3y=105$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6\times 2y=-120$$

Whakareatia te -6 ki te 2, ka -12.

$$5x-12y=-120$$

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

$$5x+3y=105,5x-12y=-120$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&3\\5&-12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}105\\-120\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&3\\5&-12\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&3\\5&-12\end{matrix}\right))\left(\begin{matrix}5&3\\5&-12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\5&-12\end{matrix}\right))\left(\begin{matrix}105\\-120\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\5&-12\end{matrix}\right))\left(\begin{matrix}105\\-120\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&3\\5&-12\end{matrix}\right))\left(\begin{matrix}105\\-120\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{12}{5\left(-12\right)-3\times 5}&-\frac{3}{5\left(-12\right)-3\times 5}\\-\frac{5}{5\left(-12\right)-3\times 5}&\frac{5}{5\left(-12\right)-3\times 5}\end{matrix}\right)\left(\begin{matrix}105\\-120\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}&\frac{1}{25}\\\frac{1}{15}&-\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}105\\-120\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{25}\times 105+\frac{1}{25}\left(-120\right)\\\frac{1}{15}\times 105-\frac{1}{15}\left(-120\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}12\\15\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=12,y=15$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 15, arā, te tauraro pātahi he tino iti rawa te kitea o 3,5.

$$5x+3y=105$$

Whakaarohia te whārite tuarua. Me whakarea ngā taha e rua o te whārite ki te 30, arā, te tauraro pātahi he tino iti rawa te kitea o 6,5.

$$5x-6\times 2y=-120$$

Whakareatia te -6 ki te 2, ka -12.

$$5x-12y=-120$$

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.

$$5x+3y=105,5x-12y=-120$$

Me tango 5x-12y=-120 mai i 5x+3y=105 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-5x+3y+12y=105+120$$

Tāpiri 5x ki te -5x. Ka whakakore atu ngā kupu 5x me -5x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$3y+12y=105+120$$

Tāpiri 3y ki te 12y.

$$15y=105+120$$

Tāpiri 105 ki te 120.

$$15y=225$$

Whakawehea ngā taha e rua ki te 15.

$$y=15$$

Whakaurua te 15 mō y ki 5x-12y=-120. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x-12\times 15=-120$$

Whakareatia -12 ki te 15.

$$5x-180=-120$$

Me tāpiri 180 ki ngā taha e rua o te whārite.

$$5x=60$$

Whakawehea ngā taha e rua ki te 5.

$$x=12$$

Kua oti te pūnaha te whakatau.

$$x=12,y=15$$