Whakaoti i te {l}{x=-1/4y+5}{3x+2y=0}

$$\left\{ \begin{array} { l } { x = - \frac { 1 } { 4 } y + 5 } \\ { 3 x + 2 y = 0 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=-12$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{4}y ki ngā taha e rua.

$$x+\frac{1}{4}y=5$$

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.

$$x+\frac{1}{4}y=5,3x+2y=0$$

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

$$x+\frac{1}{4}y=5$$

Me tango \frac{y}{4} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{4}y+5$$

Whakakapia te -\frac{y}{4}+5 mō te x ki tērā atu whārite, 3x+2y=0.

$$3\left(-\frac{1}{4}y+5\right)+2y=0$$

Whakareatia 3 ki te -\frac{y}{4}+5.

$$-\frac{3}{4}y+15+2y=0$$

Tāpiri -\frac{3y}{4} ki te 2y.

$$\frac{5}{4}y+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$\frac{5}{4}y=-15$$

Whakawehea ngā taha e rua o te whārite ki te \frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-12$$

Whakaurua te -12 mō y ki x=-\frac{1}{4}y+5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{4}\left(-12\right)+5$$

Whakareatia -\frac{1}{4} ki te -12.

$$x=3+5$$

Tāpiri 5 ki te 3.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=-12$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{4}y ki ngā taha e rua.

$$x+\frac{1}{4}y=5$$

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

$$x+\frac{1}{4}y=5,3x+2y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&\frac{1}{4}\\3&2\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5}&-\frac{1}{5}\\-\frac{12}{5}&\frac{4}{5}\end{matrix}\right)\left(\begin{matrix}5\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{5}\times 5\\-\frac{12}{5}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\-12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=-12$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me tāpiri te \frac{1}{4}y ki ngā taha e rua.

$$x+\frac{1}{4}y=5$$

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.

$$x+\frac{1}{4}y=5,3x+2y=0$$

Kia ōrite ai a x me 3x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 3 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 1.

$$3x+3\times \frac{1}{4}y=3\times 5,3x+2y=0$$

Whakarūnātia.

$$3x+\frac{3}{4}y=15,3x+2y=0$$

Me tango 3x+2y=0 mai i 3x+\frac{3}{4}y=15 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-3x+\frac{3}{4}y-2y=15$$

Tāpiri 3x ki te -3x. Ka whakakore atu ngā kupu 3x me -3x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$\frac{3}{4}y-2y=15$$

Tāpiri \frac{3y}{4} ki te -2y.

$$-\frac{5}{4}y=15$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{5}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=-12$$

Whakaurua te -12 mō y ki 3x+2y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$3x+2\left(-12\right)=0$$

Whakareatia 2 ki te -12.

$$3x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$3x=24$$

Whakawehea ngā taha e rua ki te 3.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=-12$$