Whakaoti i te {c}{x+y=122}{(y/2-11)-(x-11/2)=18}

$$\left. \begin{array} { c } { x + y = 122 } \\ { ( \frac { y } { 2 } - 11 ) - ( \frac { x - 11 } { 2 } ) = 18 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{75}{2} = 37\frac{1}{2} = 37.5$$

$$y = \frac{169}{2} = 84\frac{1}{2} = 84.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$y-22-\left(x-11\right)=36$$

Hei kimi i te tauaro o x-11, kimihia te tauaro o ia taurangi.

$$y-22-x+11=36$$

Tāpirihia te -22 ki te 11, ka -11.

$$y-11-x=36$$

Me tāpiri te 11 ki ngā taha e rua.

$$y-x=36+11$$

Tāpirihia te 36 ki te 11, ka 47.

$$y-x=47$$

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.

$$x+y=122,-x+y=47$$

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

$$x+y=122$$

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

$$x=-y+122$$

Whakakapia te -y+122 mō te x ki tērā atu whārite, -x+y=47.

$$-\left(-y+122\right)+y=47$$

Whakareatia -1 ki te -y+122.

$$y-122+y=47$$

Tāpiri y ki te y.

$$2y-122=47$$

Me tāpiri 122 ki ngā taha e rua o te whārite.

$$2y=169$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{169}{2}$$

Whakaurua te \frac{169}{2} mō y ki x=-y+122. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{169}{2}+122$$

Tāpiri 122 ki te -\frac{169}{2}.

$$x=\frac{75}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{75}{2},y=\frac{169}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$y-22-\left(x-11\right)=36$$

Hei kimi i te tauaro o x-11, kimihia te tauaro o ia taurangi.

$$y-22-x+11=36$$

Tāpirihia te -22 ki te 11, ka -11.

$$y-11-x=36$$

Me tāpiri te 11 ki ngā taha e rua.

$$y-x=36+11$$

Tāpirihia te 36 ki te 11, ka 47.

$$y-x=47$$

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

$$x+y=122,-x+y=47$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}122\\47\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}1&1\\-1&1\end{matrix}\right))\left(\begin{matrix}1&1\\-1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&1\end{matrix}\right))\left(\begin{matrix}122\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&1\end{matrix}\right))\left(\begin{matrix}122\\47\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\-1&1\end{matrix}\right))\left(\begin{matrix}122\\47\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{1-\left(-1\right)}&-\frac{1}{1-\left(-1\right)}\\-\frac{-1}{1-\left(-1\right)}&\frac{1}{1-\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}122\\47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}122\\47\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 122-\frac{1}{2}\times 47\\\frac{1}{2}\times 122+\frac{1}{2}\times 47\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{75}{2}\\\frac{169}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{75}{2},y=\frac{169}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 2.

$$y-22-\left(x-11\right)=36$$

Hei kimi i te tauaro o x-11, kimihia te tauaro o ia taurangi.

$$y-22-x+11=36$$

Tāpirihia te -22 ki te 11, ka -11.

$$y-11-x=36$$

Me tāpiri te 11 ki ngā taha e rua.

$$y-x=36+11$$

Tāpirihia te 36 ki te 11, ka 47.

$$y-x=47$$

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.

$$x+y=122,-x+y=47$$

Me tango -x+y=47 mai i x+y=122 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x+x+y-y=122-47$$

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.

$$x+x=122-47$$

Tāpiri x ki te x.

$$2x=122-47$$

Tāpiri 122 ki te -47.

$$2x=75$$

Whakawehea ngā taha e rua ki te 2.

$$x=\frac{75}{2}$$

Whakaurua te \frac{75}{2} mō x ki -x+y=47. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$-\frac{75}{2}+y=47$$

Me tāpiri \frac{75}{2} ki ngā taha e rua o te whārite.

$$y=\frac{169}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{75}{2},y=\frac{169}{2}$$