Whakaoti i te x/5=y/4;y/3=x/3-1

$$\frac { x } { 5 } = \frac { y } { 4 } ; \frac { y } { 3 } = \frac { x } { 3 } - 1$$

Whakaoti mō x, y

$$x=15$$

$$y=12$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,4.

$$4x=5y$$

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\times 5y$$

Whakareatia \frac{1}{4} ki te 5y.

$$x=\frac{5}{4}y$$

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

$$-\frac{5}{4}y+y=-3$$

Tāpiri -\frac{5y}{4} ki te y.

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

Me whakarea ngā taha e rua ki te -4.

$$y=12$$

Whakaurua te 12 mō y ki x=\frac{5}{4}y. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=\frac{5}{4}\times 12$$

Whakareatia \frac{5}{4} ki te 12.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=12$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,4.

$$4x=5y$$

Tangohia te 5y mai i ngā taha e rua.

$$4x-5y=0$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 3.

$$y=x-3$$

Tangohia te x mai i ngā taha e rua.

$$y-x=-3$$

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

$$4x-5y=0,-x+y=-3$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&-5\\-1&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}15\\12\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=15,y=12$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Me whakarea ngā taha e rua o te whārite ki te 20, arā, te tauraro pātahi he tino iti rawa te kitea o 5,4.

$$4x=5y$$

Tangohia te 5y mai i ngā taha e rua.

$$4x-5y=0$$

Whakaarohia te whārite tuarua. Whakareatia ngā taha e rua o te whārite ki te 3.

$$y=x-3$$

Tangohia te x mai i ngā taha e rua.

$$y-x=-3$$

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.

$$4x-5y=0,-x+y=-3$$

Kia ōrite ai a 4x me -x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te -1 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$-4x-\left(-5y\right)=0,4\left(-1\right)x+4y=4\left(-3\right)$$

Whakarūnātia.

$$-4x+5y=0,-4x+4y=-12$$

Me tango -4x+4y=-12 mai i -4x+5y=0 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-4x+4x+5y-4y=12$$

Tāpiri -4x ki te 4x. Ka whakakore atu ngā kupu -4x me 4x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$5y-4y=12$$

Tāpiri 5y ki te -4y.

$$y=12$$

Whakaurua te 12 mō y ki -x+y=-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+12=-3$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-x=-15$$

Whakawehea ngā taha e rua ki te -1.

$$x=15$$

Kua oti te pūnaha te whakatau.

$$x=15,y=12$$