Whakaoti i te {l}{-3x-y=2x-1}{-3(2x+5y)=x+y-30}

$$\left. \begin{array} { l } { - 3 x - y = 2 x - 1 } \\ { - 3 ( 2 x + 5 y ) = x + y - 30 } \end{array} \right.$$

Whakaoti mō x, y

$$x=-\frac{14}{73}\approx -0.191780822$$

$$y = \frac{143}{73} = 1\frac{70}{73} \approx 1.95890411$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$-3x-y-2x=-1$$

Pahekotia te -3x me -2x, ka -5x.

$$-5x-y=-1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+5y.

$$-6x-15y=x+y-30$$

Tangohia te x mai i ngā taha e rua.

$$-6x-15y-x=y-30$$

Pahekotia te -6x me -x, ka -7x.

$$-7x-15y=y-30$$

Tangohia te y mai i ngā taha e rua.

$$-7x-15y-y=-30$$

Pahekotia te -15y me -y, ka -16y.

$$-7x-16y=-30$$

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.

$$-5x-y=-1,-7x-16y=-30$$

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

$$-5x-y=-1$$

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

$$-5x=y-1$$

Whakawehea ngā taha e rua ki te -5.

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

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

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

Whakakapia te \frac{-y+1}{5} mō te x ki tērā atu whārite, -7x-16y=-30.

$$-7\left(-\frac{1}{5}y+\frac{1}{5}\right)-16y=-30$$

Whakareatia -7 ki te \frac{-y+1}{5}.

$$\frac{7}{5}y-\frac{7}{5}-16y=-30$$

Tāpiri \frac{7y}{5} ki te -16y.

$$-\frac{73}{5}y-\frac{7}{5}=-30$$

Me tāpiri \frac{7}{5} ki ngā taha e rua o te whārite.

$$-\frac{73}{5}y=-\frac{143}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{73}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{143}{73}$$

Whakaurua te \frac{143}{73} mō y ki x=-\frac{1}{5}y+\frac{1}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{5}\times \frac{143}{73}+\frac{1}{5}$$

Whakareatia -\frac{1}{5} ki te \frac{143}{73} 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.

$$x=-\frac{143}{365}+\frac{1}{5}$$

Tāpiri \frac{1}{5} ki te -\frac{143}{365} 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.

$$x=-\frac{14}{73}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{14}{73},y=\frac{143}{73}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$-3x-y-2x=-1$$

Pahekotia te -3x me -2x, ka -5x.

$$-5x-y=-1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+5y.

$$-6x-15y=x+y-30$$

Tangohia te x mai i ngā taha e rua.

$$-6x-15y-x=y-30$$

Pahekotia te -6x me -x, ka -7x.

$$-7x-15y=y-30$$

Tangohia te y mai i ngā taha e rua.

$$-7x-15y-y=-30$$

Pahekotia te -15y me -y, ka -16y.

$$-7x-16y=-30$$

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

$$-5x-y=-1,-7x-16y=-30$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right))\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right))\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right))\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&-1\\-7&-16\end{matrix}\right))\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{-5\left(-16\right)-\left(-\left(-7\right)\right)}&-\frac{-1}{-5\left(-16\right)-\left(-\left(-7\right)\right)}\\-\frac{-7}{-5\left(-16\right)-\left(-\left(-7\right)\right)}&-\frac{5}{-5\left(-16\right)-\left(-\left(-7\right)\right)}\end{matrix}\right)\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{73}&\frac{1}{73}\\\frac{7}{73}&-\frac{5}{73}\end{matrix}\right)\left(\begin{matrix}-1\\-30\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{73}\left(-1\right)+\frac{1}{73}\left(-30\right)\\\frac{7}{73}\left(-1\right)-\frac{5}{73}\left(-30\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{14}{73}\\\frac{143}{73}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{14}{73},y=\frac{143}{73}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 2x mai i ngā taha e rua.

$$-3x-y-2x=-1$$

Pahekotia te -3x me -2x, ka -5x.

$$-5x-y=-1$$

Whakaarohia te whārite tuarua. Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te 2x+5y.

$$-6x-15y=x+y-30$$

Tangohia te x mai i ngā taha e rua.

$$-6x-15y-x=y-30$$

Pahekotia te -6x me -x, ka -7x.

$$-7x-15y=y-30$$

Tangohia te y mai i ngā taha e rua.

$$-7x-15y-y=-30$$

Pahekotia te -15y me -y, ka -16y.

$$-7x-16y=-30$$

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.

$$-5x-y=-1,-7x-16y=-30$$

Kia ōrite ai a -5x 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 -5.

$$-7\left(-5\right)x-7\left(-1\right)y=-7\left(-1\right),-5\left(-7\right)x-5\left(-16\right)y=-5\left(-30\right)$$

Whakarūnātia.

$$35x+7y=7,35x+80y=150$$

Me tango 35x+80y=150 mai i 35x+7y=7 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x+7y-80y=7-150$$

Tāpiri 35x ki te -35x. Ka whakakore atu ngā kupu 35x me -35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$7y-80y=7-150$$

Tāpiri 7y ki te -80y.

$$-73y=7-150$$

Tāpiri 7 ki te -150.

$$-73y=-143$$

Whakawehea ngā taha e rua ki te -73.

$$y=\frac{143}{73}$$

Whakaurua te \frac{143}{73} mō y ki -7x-16y=-30. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x-16\times \frac{143}{73}=-30$$

Whakareatia -16 ki te \frac{143}{73}.

$$-7x-\frac{2288}{73}=-30$$

Me tāpiri \frac{2288}{73} ki ngā taha e rua o te whārite.

$$-7x=\frac{98}{73}$$

Whakawehea ngā taha e rua ki te -7.

$$x=-\frac{14}{73}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{14}{73},y=\frac{143}{73}$$