Whakaoti i te {c}{y-3x+15=10}{6-4x=y}

$$\left. \begin{array} { c } { y - 3 x + 15 = 10 } \\ { 6 - 4 x = y } \end{array} \right.$$

Whakaoti mō y, x

$$x = \frac{11}{7} = 1\frac{4}{7} \approx 1.571428571$$

$$y=-\frac{2}{7}\approx -0.285714286$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 15 mai i ngā taha e rua.

$$y-3x=10-15$$

Tangohia te 15 i te 10, ka -5.

$$y-3x=-5$$

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

$$6-4x-y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-4x-y=-6$$

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.

$$y-3x=-5,-y-4x=-6$$

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

$$y-3x=-5$$

Me tāpiri 3x ki ngā taha e rua o te whārite.

$$y=3x-5$$

Whakakapia te 3x-5 mō te y ki tērā atu whārite, -y-4x=-6.

$$-\left(3x-5\right)-4x=-6$$

Whakareatia -1 ki te 3x-5.

$$-3x+5-4x=-6$$

Tāpiri -3x ki te -4x.

$$-7x+5=-6$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-7x=-11$$

Whakawehea ngā taha e rua ki te -7.

$$x=\frac{11}{7}$$

Whakaurua te \frac{11}{7} mō x ki y=3x-5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$y=3\times \frac{11}{7}-5$$

Whakareatia 3 ki te \frac{11}{7}.

$$y=\frac{33}{7}-5$$

Tāpiri -5 ki te \frac{33}{7}.

$$y=-\frac{2}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{2}{7},x=\frac{11}{7}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 15 mai i ngā taha e rua.

$$y-3x=10-15$$

Tangohia te 15 i te 10, ka -5.

$$y-3x=-5$$

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

$$6-4x-y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-4x-y=-6$$

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

$$y-3x=-5,-y-4x=-6$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-3\\-1&-4\end{matrix}\right))\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{-4-\left(-3\left(-1\right)\right)}&-\frac{-3}{-4-\left(-3\left(-1\right)\right)}\\-\frac{-1}{-4-\left(-3\left(-1\right)\right)}&\frac{1}{-4-\left(-3\left(-1\right)\right)}\end{matrix}\right)\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}&-\frac{3}{7}\\-\frac{1}{7}&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}-5\\-6\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{4}{7}\left(-5\right)-\frac{3}{7}\left(-6\right)\\-\frac{1}{7}\left(-5\right)-\frac{1}{7}\left(-6\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{7}\\\frac{11}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa y me x.

$$y=-\frac{2}{7},x=\frac{11}{7}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 15 mai i ngā taha e rua.

$$y-3x=10-15$$

Tangohia te 15 i te 10, ka -5.

$$y-3x=-5$$

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

$$6-4x-y=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-4x-y=-6$$

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.

$$y-3x=-5,-y-4x=-6$$

Kia ōrite ai a y me -y, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$-y-\left(-3x\right)=-\left(-5\right),-y-4x=-6$$

Whakarūnātia.

$$-y+3x=5,-y-4x=-6$$

Me tango -y-4x=-6 mai i -y+3x=5 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-y+y+3x+4x=5+6$$

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.

$$3x+4x=5+6$$

Tāpiri 3x ki te 4x.

$$7x=5+6$$

Tāpiri 5 ki te 6.

$$7x=11$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{11}{7}$$

Whakaurua te \frac{11}{7} mō x ki -y-4x=-6. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-y-4\times \frac{11}{7}=-6$$

Whakareatia -4 ki te \frac{11}{7}.

$$-y-\frac{44}{7}=-6$$

Me tāpiri \frac{44}{7} ki ngā taha e rua o te whārite.

$$-y=\frac{2}{7}$$

Whakawehea ngā taha e rua ki te -1.

$$y=-\frac{2}{7}$$

Kua oti te pūnaha te whakatau.

$$y=-\frac{2}{7},x=\frac{11}{7}$$