Whakaoti i te {l}{x-6=y-6/5}{x+9=y/2}

$$\left. \begin{array} { l } { x - 6 = \frac { y - 6 } { 5 } } \\ { x + 9 = \frac { y } { 2 } } \end{array} \right.$$

Whakaoti mō x, y

$$x=14$$

$$y=46$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$5x-30=y-6$$

Tangohia te y mai i ngā taha e rua.

$$5x-30-y=-6$$

Me tāpiri te 30 ki ngā taha e rua.

$$5x-y=-6+30$$

Tāpirihia te -6 ki te 30, ka 24.

$$5x-y=24$$

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

$$2x+18=y$$

Tangohia te y mai i ngā taha e rua.

$$2x+18-y=0$$

Tangohia te 18 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-18$$

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.

$$5x-y=24,2x-y=-18$$

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

$$5x-y=24$$

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

$$5x=y+24$$

Whakawehea ngā taha e rua ki te 5.

$$x=\frac{1}{5}\left(y+24\right)$$

Whakareatia \frac{1}{5} ki te y+24.

$$x=\frac{1}{5}y+\frac{24}{5}$$

Whakakapia te \frac{24+y}{5} mō te x ki tērā atu whārite, 2x-y=-18.

$$2\left(\frac{1}{5}y+\frac{24}{5}\right)-y=-18$$

Whakareatia 2 ki te \frac{24+y}{5}.

$$\frac{2}{5}y+\frac{48}{5}-y=-18$$

Tāpiri \frac{2y}{5} ki te -y.

$$-\frac{3}{5}y+\frac{48}{5}=-18$$

Me tango \frac{48}{5} mai i ngā taha e rua o te whārite.

$$-\frac{3}{5}y=-\frac{138}{5}$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{3}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=46$$

Whakaurua te 46 mō y ki x=\frac{1}{5}y+\frac{24}{5}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{5}\times 46+\frac{24}{5}$$

Whakareatia \frac{1}{5} ki te 46.

$$x=\frac{46+24}{5}$$

Tāpiri \frac{24}{5} ki te \frac{46}{5} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=14$$

Kua oti te pūnaha te whakatau.

$$x=14,y=46$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$5x-30=y-6$$

Tangohia te y mai i ngā taha e rua.

$$5x-30-y=-6$$

Me tāpiri te 30 ki ngā taha e rua.

$$5x-y=-6+30$$

Tāpirihia te -6 ki te 30, ka 24.

$$5x-y=24$$

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

$$2x+18=y$$

Tangohia te y mai i ngā taha e rua.

$$2x+18-y=0$$

Tangohia te 18 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-18$$

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

$$5x-y=24,2x-y=-18$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}24\\-18\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}5&-1\\2&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right))\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right))\left(\begin{matrix}24\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right))\left(\begin{matrix}24\\-18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&-1\\2&-1\end{matrix}\right))\left(\begin{matrix}24\\-18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{5\left(-1\right)-\left(-2\right)}&-\frac{-1}{5\left(-1\right)-\left(-2\right)}\\-\frac{2}{5\left(-1\right)-\left(-2\right)}&\frac{5}{5\left(-1\right)-\left(-2\right)}\end{matrix}\right)\left(\begin{matrix}24\\-18\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}&-\frac{1}{3}\\\frac{2}{3}&-\frac{5}{3}\end{matrix}\right)\left(\begin{matrix}24\\-18\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{3}\times 24-\frac{1}{3}\left(-18\right)\\\frac{2}{3}\times 24-\frac{5}{3}\left(-18\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14\\46\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=14,y=46$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakareatia ngā taha e rua o te whārite ki te 5.

$$5x-30=y-6$$

Tangohia te y mai i ngā taha e rua.

$$5x-30-y=-6$$

Me tāpiri te 30 ki ngā taha e rua.

$$5x-y=-6+30$$

Tāpirihia te -6 ki te 30, ka 24.

$$5x-y=24$$

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

$$2x+18=y$$

Tangohia te y mai i ngā taha e rua.

$$2x+18-y=0$$

Tangohia te 18 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-y=-18$$

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.

$$5x-y=24,2x-y=-18$$

Me tango 2x-y=-18 mai i 5x-y=24 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$5x-2x-y+y=24+18$$

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.

$$5x-2x=24+18$$

Tāpiri 5x ki te -2x.

$$3x=24+18$$

Tāpiri 24 ki te 18.

$$3x=42$$

Whakawehea ngā taha e rua ki te 3.

$$x=14$$

Whakaurua te 14 mō x ki 2x-y=-18. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$2\times 14-y=-18$$

Whakareatia 2 ki te 14.

$$28-y=-18$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$-y=-46$$

Whakawehea ngā taha e rua ki te -1.

$$y=46$$

Kua oti te pūnaha te whakatau.

$$x=14,y=46$$