Whakaoti i te {l}{x/5-2=y/10}{5x+45=-7y}

$$\left. \begin{array} { l } { \frac { x } { 5 } - 2 = \frac { y } { 10 } } \\ { 5 x + 45 = - 7 y } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=-10$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,10.

$$2x-20=y$$

Tangohia te y mai i ngā taha e rua.

$$2x-20-y=0$$

Me tāpiri te 20 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-y=20$$

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x+45+7y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x+7y=-45$$

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.

$$2x-y=20,5x+7y=-45$$

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

$$2x-y=20$$

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

$$2x=y+20$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{1}{2}\left(y+20\right)$$

Whakareatia \frac{1}{2} ki te y+20.

$$x=\frac{1}{2}y+10$$

Whakakapia te \frac{y}{2}+10 mō te x ki tērā atu whārite, 5x+7y=-45.

$$5\left(\frac{1}{2}y+10\right)+7y=-45$$

Whakareatia 5 ki te \frac{y}{2}+10.

$$\frac{5}{2}y+50+7y=-45$$

Tāpiri \frac{5y}{2} ki te 7y.

$$\frac{19}{2}y+50=-45$$

Me tango 50 mai i ngā taha e rua o te whārite.

$$\frac{19}{2}y=-95$$

Whakawehea ngā taha e rua o te whārite ki te \frac{19}{2}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-10$$

Whakaurua te -10 mō y ki x=\frac{1}{2}y+10. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{2}\left(-10\right)+10$$

Whakareatia \frac{1}{2} ki te -10.

$$x=-5+10$$

Tāpiri 10 ki te -5.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-10$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,10.

$$2x-20=y$$

Tangohia te y mai i ngā taha e rua.

$$2x-20-y=0$$

Me tāpiri te 20 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-y=20$$

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x+45+7y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x+7y=-45$$

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

$$2x-y=20,5x+7y=-45$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}2&-1\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\-45\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}2&-1\\5&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-1\\5&7\end{matrix}\right))\left(\begin{matrix}2&-1\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\5&7\end{matrix}\right))\left(\begin{matrix}20\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\5&7\end{matrix}\right))\left(\begin{matrix}20\\-45\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&-1\\5&7\end{matrix}\right))\left(\begin{matrix}20\\-45\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{2\times 7-\left(-5\right)}&-\frac{-1}{2\times 7-\left(-5\right)}\\-\frac{5}{2\times 7-\left(-5\right)}&\frac{2}{2\times 7-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}20\\-45\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{19}&\frac{1}{19}\\-\frac{5}{19}&\frac{2}{19}\end{matrix}\right)\left(\begin{matrix}20\\-45\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{19}\times 20+\frac{1}{19}\left(-45\right)\\-\frac{5}{19}\times 20+\frac{2}{19}\left(-45\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa x me y.

$$x=5,y=-10$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 10, arā, te tauraro pātahi he tino iti rawa te kitea o 5,10.

$$2x-20=y$$

Tangohia te y mai i ngā taha e rua.

$$2x-20-y=0$$

Me tāpiri te 20 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2x-y=20$$

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x+45+7y=0$$

Tangohia te 45 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$5x+7y=-45$$

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.

$$2x-y=20,5x+7y=-45$$

Kia ōrite ai a 2x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 2.

$$5\times 2x+5\left(-1\right)y=5\times 20,2\times 5x+2\times 7y=2\left(-45\right)$$

Whakarūnātia.

$$10x-5y=100,10x+14y=-90$$

Me tango 10x+14y=-90 mai i 10x-5y=100 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$10x-10x-5y-14y=100+90$$

Tāpiri 10x ki te -10x. Ka whakakore atu ngā kupu 10x me -10x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-5y-14y=100+90$$

Tāpiri -5y ki te -14y.

$$-19y=100+90$$

Tāpiri 100 ki te 90.

$$-19y=190$$

Whakawehea ngā taha e rua ki te -19.

$$y=-10$$

Whakaurua te -10 mō y ki 5x+7y=-45. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+7\left(-10\right)=-45$$

Whakareatia 7 ki te -10.

$$5x-70=-45$$

Me tāpiri 70 ki ngā taha e rua o te whārite.

$$5x=25$$

Whakawehea ngā taha e rua ki te 5.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=-10$$