Whakaoti i te {l}{y=5x+3}{y=2x+b}

$$\left. \begin{array} { l } { y = 5 x + 3 } \\ { y = 2 x + b } \end{array} \right.$$

Whakaoti mō x, y

$$x=\frac{b}{3}-1$$

$$y=\frac{5b}{3}-2$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=b$$

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.

$$y-5x=3,y-2x=b$$

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

$$y-5x=3$$

Me tāpiri 5x ki ngā taha e rua o te whārite.

$$y=5x+3$$

Whakakapia te 5x+3 mō te y ki tērā atu whārite, y-2x=b.

$$5x+3-2x=b$$

Tāpiri 5x ki te -2x.

$$3x+3=b$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$3x=b-3$$

Whakawehea ngā taha e rua ki te 3.

$$x=\frac{b}{3}-1$$

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

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

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

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

Tāpiri 3 ki te -5+\frac{5b}{3}.

$$y=\frac{5b}{3}-2$$

Kua oti te pūnaha te whakatau.

$$y=\frac{5b}{3}-2,x=\frac{b}{3}-1$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=b$$

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

$$y-5x=3,y-2x=b$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

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

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}1&-5\\1&-2\end{matrix}\right).

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

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-5\\1&-2\end{matrix}\right))\left(\begin{matrix}3\\b\end{matrix}\right)$$

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

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

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.

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

Mahia ngā tātaitanga.

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

Whakareatia ngā poukapa.

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

Mahia ngā tātaitanga.

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

Tangohia ngā huānga poukapa y me x.

$$y=\frac{5b}{3}-2,x=\frac{b}{3}-1$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuatahi. Tangohia te 5x mai i ngā taha e rua.

$$y-5x=3$$

Whakaarohia te whārite tuarua. Tangohia te 2x mai i ngā taha e rua.

$$y-2x=b$$

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.

$$y-5x=3,y-2x=b$$

Me tango y-2x=b mai i y-5x=3 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$y-y-5x+2x=3-b$$

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.

$$-5x+2x=3-b$$

Tāpiri -5x ki te 2x.

$$-3x=3-b$$

Whakawehea ngā taha e rua ki te -3.

$$x=\frac{b}{3}-1$$

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

$$y-2\left(\frac{b}{3}-1\right)=b$$

Whakareatia -2 ki te -1+\frac{b}{3}.

$$y-\frac{2b}{3}+2=b$$

Me tango 2-\frac{2b}{3} mai i ngā taha e rua o te whārite.

$$y=\frac{5b}{3}-2$$

Kua oti te pūnaha te whakatau.

$$y=\frac{5b}{3}-2,x=\frac{b}{3}-1$$