Whakaoti i te {l}{-5x+8y=0}{-7x-8y=-96}

$$\left. \begin{array} { l } { - 5 x + 8 y = 0 } \\ { - 7 x - 8 y = - 96 } \end{array} \right.$$

Whakaoti mō x, y

$$x=8$$

$$y=5$$

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.

$$-5x+8y=0,-7x-8y=-96$$

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

$$-5x+8y=0$$

Me tango 8y mai i ngā taha e rua o te whārite.

$$-5x=-8y$$

Whakawehea ngā taha e rua ki te -5.

$$x=-\frac{1}{5}\left(-8\right)y$$

Whakareatia -\frac{1}{5} ki te -8y.

$$x=\frac{8}{5}y$$

Whakakapia te \frac{8y}{5} mō te x ki tērā atu whārite, -7x-8y=-96.

$$-7\times \frac{8}{5}y-8y=-96$$

Whakareatia -7 ki te \frac{8y}{5}.

$$-\frac{56}{5}y-8y=-96$$

Tāpiri -\frac{56y}{5} ki te -8y.

$$-\frac{96}{5}y=-96$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{96}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=5$$

Whakaurua te 5 mō y ki x=\frac{8}{5}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{8}{5}\times 5$$

Whakareatia \frac{8}{5} ki te 5.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=5$$

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.

$$-5x+8y=0,-7x-8y=-96$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\-96\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right))\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right))\left(\begin{matrix}0\\-96\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right))\left(\begin{matrix}0\\-96\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5&8\\-7&-8\end{matrix}\right))\left(\begin{matrix}0\\-96\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8}{-5\left(-8\right)-8\left(-7\right)}&-\frac{8}{-5\left(-8\right)-8\left(-7\right)}\\-\frac{-7}{-5\left(-8\right)-8\left(-7\right)}&-\frac{5}{-5\left(-8\right)-8\left(-7\right)}\end{matrix}\right)\left(\begin{matrix}0\\-96\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}&-\frac{1}{12}\\\frac{7}{96}&-\frac{5}{96}\end{matrix}\right)\left(\begin{matrix}0\\-96\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{12}\left(-96\right)\\-\frac{5}{96}\left(-96\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\5\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=8,y=5$$

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.

$$-5x+8y=0,-7x-8y=-96$$

Kia ōrite ai a -5x me -7x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -7 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te -5.

$$-7\left(-5\right)x-7\times 8y=0,-5\left(-7\right)x-5\left(-8\right)y=-5\left(-96\right)$$

Whakarūnātia.

$$35x-56y=0,35x+40y=480$$

Me tango 35x+40y=480 mai i 35x-56y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$35x-35x-56y-40y=-480$$

Tāpiri 35x ki te -35x. Ka whakakore atu ngā kupu 35x me -35x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$-56y-40y=-480$$

Tāpiri -56y ki te -40y.

$$-96y=-480$$

Whakawehea ngā taha e rua ki te -96.

$$y=5$$

Whakaurua te 5 mō y ki -7x-8y=-96. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-7x-8\times 5=-96$$

Whakareatia -8 ki te 5.

$$-7x-40=-96$$

Me tāpiri 40 ki ngā taha e rua o te whārite.

$$-7x=-56$$

Whakawehea ngā taha e rua ki te -7.

$$x=8$$

Kua oti te pūnaha te whakatau.

$$x=8,y=5$$