Whakaoti i te {l}{8x+y=64}{x+y=42}

$$\left. \begin{array} { l } { 8 x + y = 64 } \\ { x + y = 42 } \end{array} \right.$$

Whakaoti mō x, y

$$x = \frac{22}{7} = 3\frac{1}{7} \approx 3.142857143$$

$$y = \frac{272}{7} = 38\frac{6}{7} \approx 38.857142857$$

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.

$$8x+y=64,x+y=42$$

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

$$8x+y=64$$

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

$$8x=-y+64$$

Whakawehea ngā taha e rua ki te 8.

$$x=\frac{1}{8}\left(-y+64\right)$$

Whakareatia \frac{1}{8} ki te -y+64.

$$x=-\frac{1}{8}y+8$$

Whakakapia te -\frac{y}{8}+8 mō te x ki tērā atu whārite, x+y=42.

$$-\frac{1}{8}y+8+y=42$$

Tāpiri -\frac{y}{8} ki te y.

$$\frac{7}{8}y+8=42$$

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

$$\frac{7}{8}y=34$$

Whakawehea ngā taha e rua o te whārite ki te \frac{7}{8}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{272}{7}$$

Whakaurua te \frac{272}{7} mō y ki x=-\frac{1}{8}y+8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{1}{8}\times \frac{272}{7}+8$$

Whakareatia -\frac{1}{8} ki te \frac{272}{7} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=-\frac{34}{7}+8$$

Tāpiri 8 ki te -\frac{34}{7}.

$$x=\frac{22}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{7},y=\frac{272}{7}$$

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.

$$8x+y=64,x+y=42$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}8&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}64\\42\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}8&1\\1&1\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&1\\1&1\end{matrix}\right))\left(\begin{matrix}8&1\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&1\end{matrix}\right))\left(\begin{matrix}64\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&1\end{matrix}\right))\left(\begin{matrix}64\\42\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&1\\1&1\end{matrix}\right))\left(\begin{matrix}64\\42\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{8-1}&-\frac{1}{8-1}\\-\frac{1}{8-1}&\frac{8}{8-1}\end{matrix}\right)\left(\begin{matrix}64\\42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}&-\frac{1}{7}\\-\frac{1}{7}&\frac{8}{7}\end{matrix}\right)\left(\begin{matrix}64\\42\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{7}\times 64-\frac{1}{7}\times 42\\-\frac{1}{7}\times 64+\frac{8}{7}\times 42\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{22}{7}\\\frac{272}{7}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=\frac{22}{7},y=\frac{272}{7}$$

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.

$$8x+y=64,x+y=42$$

Me tango x+y=42 mai i 8x+y=64 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$8x-x+y-y=64-42$$

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.

$$8x-x=64-42$$

Tāpiri 8x ki te -x.

$$7x=64-42$$

Tāpiri 64 ki te -42.

$$7x=22$$

Whakawehea ngā taha e rua ki te 7.

$$x=\frac{22}{7}$$

Whakaurua te \frac{22}{7} mō x ki x+y=42. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō y hāngai tonu.

$$\frac{22}{7}+y=42$$

Me tango \frac{22}{7} mai i ngā taha e rua o te whārite.

$$y=\frac{272}{7}$$

Kua oti te pūnaha te whakatau.

$$x=\frac{22}{7},y=\frac{272}{7}$$