Whakaoti i te {l}{x+y=40}{x=1/7y}

$$\left\{ \begin{array} { l } { x + y = 40 } \\ { x = \frac { 1 } { 7 } y } \end{array} \right.$$

Whakaoti mō x, y

$$x=5$$

$$y=35$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{7}y mai i ngā taha e rua.

$$x-\frac{1}{7}y=0$$

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.

$$x+y=40,x-\frac{1}{7}y=0$$

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

$$x+y=40$$

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

$$x=-y+40$$

Whakakapia te -y+40 mō te x ki tērā atu whārite, x-\frac{1}{7}y=0.

$$-y+40-\frac{1}{7}y=0$$

Tāpiri -y ki te -\frac{y}{7}.

$$-\frac{8}{7}y+40=0$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$-\frac{8}{7}y=-40$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{8}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=35$$

Whakaurua te 35 mō y ki x=-y+40. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-35+40$$

Tāpiri 40 ki te -35.

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=35$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{7}y mai i ngā taha e rua.

$$x-\frac{1}{7}y=0$$

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

$$x+y=40,x-\frac{1}{7}y=0$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\0\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right))\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right))\left(\begin{matrix}40\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right))\left(\begin{matrix}40\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1&-\frac{1}{7}\end{matrix}\right))\left(\begin{matrix}40\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{\frac{1}{7}}{-\frac{1}{7}-1}&-\frac{1}{-\frac{1}{7}-1}\\-\frac{1}{-\frac{1}{7}-1}&\frac{1}{-\frac{1}{7}-1}\end{matrix}\right)\left(\begin{matrix}40\\0\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}&\frac{7}{8}\\\frac{7}{8}&-\frac{7}{8}\end{matrix}\right)\left(\begin{matrix}40\\0\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8}\times 40\\\frac{7}{8}\times 40\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}5\\35\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=5,y=35$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Tangohia te \frac{1}{7}y mai i ngā taha e rua.

$$x-\frac{1}{7}y=0$$

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.

$$x+y=40,x-\frac{1}{7}y=0$$

Me tango x-\frac{1}{7}y=0 mai i x+y=40 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$x-x+y+\frac{1}{7}y=40$$

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.

$$y+\frac{1}{7}y=40$$

Tāpiri y ki te \frac{y}{7}.

$$\frac{8}{7}y=40$$

Whakawehea ngā taha e rua o te whārite ki te \frac{8}{7}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=35$$

Whakaurua te 35 mō y ki x-\frac{1}{7}y=0. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x-\frac{1}{7}\times 35=0$$

Whakareatia -\frac{1}{7} ki te 35.

$$x-5=0$$

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

$$x=5$$

Kua oti te pūnaha te whakatau.

$$x=5,y=35$$