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

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

Whakaoti mō y, x

$$x=-1$$

$$y=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=4$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=-3$$

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.

$$y-5x=4,y+2x=-3$$

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

$$y-5x=4$$

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

$$y=5x+4$$

Whakakapia te 5x+4 mō te y ki tērā atu whārite, y+2x=-3.

$$5x+4+2x=-3$$

Tāpiri 5x ki te 2x.

$$7x+4=-3$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$7x=-7$$

Whakawehea ngā taha e rua ki te 7.

$$x=-1$$

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

$$y=5\left(-1\right)+4$$

Whakareatia 5 ki te -1.

$$y=-5+4$$

Tāpiri 4 ki te -5.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=4$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=-3$$

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

$$y-5x=4,y+2x=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-5\\1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}4\\-3\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&-5\\1&2\end{matrix}\right))\left(\begin{matrix}1&-5\\1&2\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&2\end{matrix}\right))\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te poukapa kōaro ko \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), kia tuhia anō ai te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{2-\left(-5\right)}&-\frac{-5}{2-\left(-5\right)}\\-\frac{1}{2-\left(-5\right)}&\frac{1}{2-\left(-5\right)}\end{matrix}\right)\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}&\frac{5}{7}\\-\frac{1}{7}&\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}4\\-3\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{2}{7}\times 4+\frac{5}{7}\left(-3\right)\\-\frac{1}{7}\times 4+\frac{1}{7}\left(-3\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=-1,x=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=4$$

Whakaarohia te whārite tuarua. Me tāpiri te 2x ki ngā taha e rua.

$$y+2x=-3$$

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.

$$y-5x=4,y+2x=-3$$

Me tango y+2x=-3 mai i y-5x=4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-5x-2x=4+3$$

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

$$-5x-2x=4+3$$

Tāpiri -5x ki te -2x.

$$-7x=4+3$$

Tāpiri 4 ki te 3.

$$-7x=7$$

Whakawehea ngā taha e rua ki te -7.

$$x=-1$$

Whakaurua te -1 mō x ki y+2x=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y+2\left(-1\right)=-3$$

Whakareatia 2 ki te -1.

$$y-2=-3$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$y=-1$$

Kua oti te pūnaha te whakatau.

$$y=-1,x=-1$$