Whakaoti i te {l}{5=3k+b}{-9=-4k+b}

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

Whakaoti mō k, b

$$k=2$$

$$b=-1$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3k+b=5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-4k+b=-9$$

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.

$$3k+b=5,-4k+b=-9$$

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

$$3k+b=5$$

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

$$3k=-b+5$$

Whakawehea ngā taha e rua ki te 3.

$$k=\frac{1}{3}\left(-b+5\right)$$

Whakareatia \frac{1}{3} ki te -b+5.

$$k=-\frac{1}{3}b+\frac{5}{3}$$

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

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

Whakareatia -4 ki te \frac{-b+5}{3}.

$$\frac{4}{3}b-\frac{20}{3}+b=-9$$

Tāpiri \frac{4b}{3} ki te b.

$$\frac{7}{3}b-\frac{20}{3}=-9$$

Me tāpiri \frac{20}{3} ki ngā taha e rua o te whārite.

$$\frac{7}{3}b=-\frac{7}{3}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{3}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$b=-1$$

Whakaurua te -1 mō b ki k=-\frac{1}{3}b+\frac{5}{3}. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō k hāngai tonu.

$$k=-\frac{1}{3}\left(-1\right)+\frac{5}{3}$$

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

$$k=\frac{1+5}{3}$$

Tāpiri \frac{5}{3} ki te \frac{1}{3} 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.

$$k=2$$

Kua oti te pūnaha te whakatau.

$$k=2,b=-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3k+b=5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-4k+b=-9$$

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

$$3k+b=5,-4k+b=-9$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}3&1\\-4&1\end{matrix}\right).

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

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

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

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

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

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.

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

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{1}{7}\\\frac{4}{7}&\frac{3}{7}\end{matrix}\right)\left(\begin{matrix}5\\-9\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 5-\frac{1}{7}\left(-9\right)\\\frac{4}{7}\times 5+\frac{3}{7}\left(-9\right)\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}k\\b\end{matrix}\right)=\left(\begin{matrix}2\\-1\end{matrix}\right)$$

Tangohia ngā huānga poukapa k me b.

$$k=2,b=-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$3k+b=5$$

Whakaarohia te whārite tuarua. Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-4k+b=-9$$

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.

$$3k+b=5,-4k+b=-9$$

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

$$3k+4k+b-b=5+9$$

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

$$3k+4k=5+9$$

Tāpiri 3k ki te 4k.

$$7k=5+9$$

Tāpiri 5 ki te 9.

$$7k=14$$

Whakawehea ngā taha e rua ki te 7.

$$k=2$$

Whakaurua te 2 mō k ki -4k+b=-9. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō b hāngai tonu.

$$-4\times 2+b=-9$$

Whakareatia -4 ki te 2.

$$-8+b=-9$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$b=-1$$

Kua oti te pūnaha te whakatau.

$$k=2,b=-1$$