Whakaoti i te {l}{x-3=y}{x/4-1=y}

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

Whakaoti mō x, y

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

$$y=-\frac{1}{3}\approx -0.333333333$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$\frac{x}{4}-1-y=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-4y=4$$

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.

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

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

$$x-y=3$$

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

$$x=y+3$$

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

$$y+3-4y=4$$

Tāpiri y ki te -4y.

$$-3y+3=4$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-3y=1$$

Whakawehea ngā taha e rua ki te -3.

$$y=-\frac{1}{3}$$

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

$$x=-\frac{1}{3}+3$$

Tāpiri 3 ki te -\frac{1}{3}.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=-\frac{1}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$\frac{x}{4}-1-y=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-4y=4$$

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

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

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

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

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{3}\\-\frac{1}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{8}{3},y=-\frac{1}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$x-3-y=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x-y=3$$

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

$$\frac{x}{4}-1-y=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakareatia ngā taha e rua o te whārite ki te 4.

$$x-4y=4$$

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.

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

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

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

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

$$-y+4y=3-4$$

Tāpiri -y ki te 4y.

$$3y=3-4$$

Tāpiri 3 ki te -4.

$$3y=-1$$

Whakawehea ngā taha e rua ki te 3.

$$y=-\frac{1}{3}$$

Whakaurua te -\frac{1}{3} mō y ki x-4y=4. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-4\left(-\frac{1}{3}\right)=4$$

Whakareatia -4 ki te -\frac{1}{3}.

$$x+\frac{4}{3}=4$$

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$x=\frac{8}{3}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{8}{3},y=-\frac{1}{3}$$