Whakaoti i te {l}{4x+5y=-12}{5x-17=-7y}

$$\left. \begin{array} { l } { 4 x + 5 y = - 12 } \\ { 5 x - 17 = - 7 y } \end{array} \right.$$

Whakaoti mō x, y

$$x = -\frac{169}{3} = -56\frac{1}{3} \approx -56.333333333$$

$$y = \frac{128}{3} = 42\frac{2}{3} \approx 42.666666667$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x-17+7y=0$$

Me tāpiri te 17 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x+7y=17$$

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.

$$4x+5y=-12,5x+7y=17$$

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

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

Me tango 5y mai i ngā taha e rua o te whārite.

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

Whakawehea ngā taha e rua ki te 4.

$$x=\frac{1}{4}\left(-5y-12\right)$$

Whakareatia \frac{1}{4} ki te -5y-12.

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

Whakakapia te -\frac{5y}{4}-3 mō te x ki tērā atu whārite, 5x+7y=17.

$$5\left(-\frac{5}{4}y-3\right)+7y=17$$

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

$$-\frac{25}{4}y-15+7y=17$$

Tāpiri -\frac{25y}{4} ki te 7y.

$$\frac{3}{4}y-15=17$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$\frac{3}{4}y=32$$

Whakawehea ngā taha e rua o te whārite ki te \frac{3}{4}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$y=\frac{128}{3}$$

Whakaurua te \frac{128}{3} mō y ki x=-\frac{5}{4}y-3. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=-\frac{5}{4}\times \frac{128}{3}-3$$

Whakareatia -\frac{5}{4} ki te \frac{128}{3} 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.

$$x=-\frac{160}{3}-3$$

Tāpiri -3 ki te -\frac{160}{3}.

$$x=-\frac{169}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{169}{3},y=\frac{128}{3}$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x-17+7y=0$$

Me tāpiri te 17 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x+7y=17$$

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

$$4x+5y=-12,5x+7y=17$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}4&5\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-12\\17\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}4&5\\5&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}4&5\\5&7\end{matrix}\right))\left(\begin{matrix}4&5\\5&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&7\end{matrix}\right))\left(\begin{matrix}-12\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&7\end{matrix}\right))\left(\begin{matrix}-12\\17\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}4&5\\5&7\end{matrix}\right))\left(\begin{matrix}-12\\17\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{4\times 7-5\times 5}&-\frac{5}{4\times 7-5\times 5}\\-\frac{5}{4\times 7-5\times 5}&\frac{4}{4\times 7-5\times 5}\end{matrix}\right)\left(\begin{matrix}-12\\17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}&-\frac{5}{3}\\-\frac{5}{3}&\frac{4}{3}\end{matrix}\right)\left(\begin{matrix}-12\\17\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{3}\left(-12\right)-\frac{5}{3}\times 17\\-\frac{5}{3}\left(-12\right)+\frac{4}{3}\times 17\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{169}{3}\\\frac{128}{3}\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=-\frac{169}{3},y=\frac{128}{3}$$

Ngā Upane e Whakamahi i te Tangohanga

Whakaarohia te whārite tuarua. Me tāpiri te 7y ki ngā taha e rua.

$$5x-17+7y=0$$

Me tāpiri te 17 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$5x+7y=17$$

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.

$$4x+5y=-12,5x+7y=17$$

Kia ōrite ai a 4x me 5x, whakareatia ngā kīanga tau katoa kei ia taha o te whārite tuatahi ki te 5 me ngā kīanga tau katoa kei ia taha o te whārite tuarua ki te 4.

$$5\times 4x+5\times 5y=5\left(-12\right),4\times 5x+4\times 7y=4\times 17$$

Whakarūnātia.

$$20x+25y=-60,20x+28y=68$$

Me tango 20x+28y=68 mai i 20x+25y=-60 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$20x-20x+25y-28y=-60-68$$

Tāpiri 20x ki te -20x. Ka whakakore atu ngā kupu 20x me -20x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$25y-28y=-60-68$$

Tāpiri 25y ki te -28y.

$$-3y=-60-68$$

Tāpiri -60 ki te -68.

$$-3y=-128$$

Whakawehea ngā taha e rua ki te -3.

$$y=\frac{128}{3}$$

Whakaurua te \frac{128}{3} mō y ki 5x+7y=17. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$5x+7\times \frac{128}{3}=17$$

Whakareatia 7 ki te \frac{128}{3}.

$$5x+\frac{896}{3}=17$$

Me tango \frac{896}{3} mai i ngā taha e rua o te whārite.

$$5x=-\frac{845}{3}$$

Whakawehea ngā taha e rua ki te 5.

$$x=-\frac{169}{3}$$

Kua oti te pūnaha te whakatau.

$$x=-\frac{169}{3},y=\frac{128}{3}$$