Whakaoti i te {l}{3x+4=y}{9x-5=y}

$$\left. \begin{array} { l } { 3 x + 4 = y } \\ { 9 x - 5 = y } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$y = \frac{17}{2} = 8\frac{1}{2} = 8.5$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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

$$3x+4-y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-y=-4$$

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

$$9x-5-y=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-y=5$$

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.

$$3x-y=-4,9x-y=5$$

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

$$3x-y=-4$$

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

$$3x=y-4$$

Whakawehea ngā taha e rua ki te 3.

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

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

$$x=\frac{1}{3}y-\frac{4}{3}$$

Whakakapia te \frac{-4+y}{3} mō te x ki tērā atu whārite, 9x-y=5.

$$9\left(\frac{1}{3}y-\frac{4}{3}\right)-y=5$$

Whakareatia 9 ki te \frac{-4+y}{3}.

$$3y-12-y=5$$

Tāpiri 3y ki te -y.

$$2y-12=5$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$2y=17$$

Whakawehea ngā taha e rua ki te 2.

$$y=\frac{17}{2}$$

Whakaurua te \frac{17}{2} mō y ki x=\frac{1}{3}y-\frac{4}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{1}{3}\times \frac{17}{2}-\frac{4}{3}$$

Whakareatia \frac{1}{3} ki te \frac{17}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{17}{6}-\frac{4}{3}$$

Tāpiri -\frac{4}{3} ki te \frac{17}{6} 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.

$$x=\frac{3}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{17}{2}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$3x+4-y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-y=-4$$

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

$$9x-5-y=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-y=5$$

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

$$3x-y=-4,9x-y=5$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-4\\5\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&-1\\9&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right))\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right))\left(\begin{matrix}-4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right))\left(\begin{matrix}-4\\5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&-1\\9&-1\end{matrix}\right))\left(\begin{matrix}-4\\5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{3\left(-1\right)-\left(-9\right)}&-\frac{-1}{3\left(-1\right)-\left(-9\right)}\\-\frac{9}{3\left(-1\right)-\left(-9\right)}&\frac{3}{3\left(-1\right)-\left(-9\right)}\end{matrix}\right)\left(\begin{matrix}-4\\5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{6}\\-\frac{3}{2}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}-4\\5\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\left(-4\right)+\frac{1}{6}\times 5\\-\frac{3}{2}\left(-4\right)+\frac{1}{2}\times 5\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2}\\\frac{17}{2}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{3}{2},y=\frac{17}{2}$$

Ngā Upane e Whakamahi i te Tangohanga

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

$$3x+4-y=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$3x-y=-4$$

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

$$9x-5-y=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-y=5$$

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.

$$3x-y=-4,9x-y=5$$

Me tango 9x-y=5 mai i 3x-y=-4 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$3x-9x-y+y=-4-5$$

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.

$$3x-9x=-4-5$$

Tāpiri 3x ki te -9x.

$$-6x=-4-5$$

Tāpiri -4 ki te -5.

$$-6x=-9$$

Whakawehea ngā taha e rua ki te -6.

$$x=\frac{3}{2}$$

Whakaurua te \frac{3}{2} mō x ki 9x-y=5. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$9\times \frac{3}{2}-y=5$$

Whakareatia 9 ki te \frac{3}{2}.

$$\frac{27}{2}-y=5$$

Me tango \frac{27}{2} mai i ngā taha e rua o te whārite.

$$-y=-\frac{17}{2}$$

Whakawehea ngā taha e rua ki te -1.

$$y=\frac{17}{2}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{3}{2},y=\frac{17}{2}$$