Whakaoti i te {l}{4x+9y=28}{-4x-y=-28}

$$\left. \begin{array} { l } { 4 x + 9 y = 28 } \\ { - 4 x - y = - 28 } \end{array} \right.$$

Whakaoti mō x, y

$$x=7$$

$$y=0$$

Ngā Upane e Whakamahi ana i te Whakakapinga

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.

$$4x+9y=28,-4x-y=-28$$

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

$$4x+9y=28$$

Me tango 9y mai i ngā taha e rua o te whārite.

$$4x=-9y+28$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-9y+28\right)$$

Whakareatia \frac{1}{4} ki te -9y+28.

$$x=-\frac{9}{4}y+7$$

Whakakapia te -\frac{9y}{4}+7 mō te x ki tērā atu whārite, -4x-y=-28.

$$-4\left(-\frac{9}{4}y+7\right)-y=-28$$

Whakareatia -4 ki te -\frac{9y}{4}+7.

$$9y-28-y=-28$$

Tāpiri 9y ki te -y.

$$8y-28=-28$$

Me tāpiri 28 ki ngā taha e rua o te whārite.

$$8y=0$$

Whakawehea ngā taha e rua ki te 8.

$$y=0$$

Whakaurua te 0 mō y ki x=-\frac{9}{4}y+7. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=0$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

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

$$4x+9y=28,-4x-y=-28$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}28\\-28\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&9\\-4&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right))\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right))\left(\begin{matrix}28\\-28\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right))\left(\begin{matrix}28\\-28\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&9\\-4&-1\end{matrix}\right))\left(\begin{matrix}28\\-28\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{4\left(-1\right)-9\left(-4\right)}&-\frac{9}{4\left(-1\right)-9\left(-4\right)}\\-\frac{-4}{4\left(-1\right)-9\left(-4\right)}&\frac{4}{4\left(-1\right)-9\left(-4\right)}\end{matrix}\right)\left(\begin{matrix}28\\-28\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{32}&-\frac{9}{32}\\\frac{1}{8}&\frac{1}{8}\end{matrix}\right)\left(\begin{matrix}28\\-28\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{32}\times 28-\frac{9}{32}\left(-28\right)\\\frac{1}{8}\times 28+\frac{1}{8}\left(-28\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}7\\0\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=7,y=0$$

Ngā Upane e Whakamahi i te Tangohanga

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.

$$4x+9y=28,-4x-y=-28$$

Kia ōrite ai a 4x me -4x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -4 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-4\times 4x-4\times 9y=-4\times 28,4\left(-4\right)x+4\left(-1\right)y=4\left(-28\right)$$

Whakarūnātia.

$$-16x-36y=-112,-16x-4y=-112$$

Me tango -16x-4y=-112 mai i -16x-36y=-112 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-16x+16x-36y+4y=-112+112$$

Tāpiri -16x ki te 16x. Ka whakakore atu ngā kupu -16x me 16x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-36y+4y=-112+112$$

Tāpiri -36y ki te 4y.

$$-32y=-112+112$$

Tāpiri -112 ki te 112.

$$-32y=0$$

Whakawehea ngā taha e rua ki te -32.

$$y=0$$

Whakaurua te 0 mō y ki -4x-y=-28. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-4x=-28$$

Whakawehea ngā taha e rua ki te -4.

$$x=7$$

Kua oti te pūnaha te whakatau.

$$x=7,y=0$$